1) Find the following limit:
a) x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\
\to\end{array}$2 (2x2 + 3x – 14)
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\ \to\end{array}$2
(2x2 + 3x – 14) = 2 * (2)2 + 3 * 2 – 14 = 0
b) x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\
\to\end{array}$5 (x2 + 2x – 9)
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\ \to\end{array}$5
(x2 + 2x – 9) = (5)2 + 2 * 5 – 9= 26
c) x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\
\to\end{array}$1 $\frac{{3{{\rm{x}}^2} + 2{\rm{x}} - 4}}{{{{\rm{x}}^2} +
5{\rm{x}} - 4}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\ \to\end{array}$1
$\frac{{3{{\rm{x}}^2} + 2{\rm{x}} - 4}}{{{{\rm{x}}^2} + 5{\rm{x}} - 4}}$ =
$\frac{{3{\rm{*}}{{\left( 1 \right)}^2} + 2{\rm{*}}1 - 4}}{{{{\left( 1
\right)}^2} + 5{\rm{*}}1 - 4}}$ = $\frac{1}{2}$.
d) x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\
\to\end{array}$3 $\frac{{6{{\rm{x}}^2} + 3{\rm{x}} - 12}}{{2{{\rm{x}}^2} +
{\rm{x}} + 1}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\ \to\end{array}$3
$\frac{{6{{\rm{x}}^2} + 3{\rm{x}} - 12}}{{2{{\rm{x}}^2} + {\rm{x}} + 1}}$ =
$\frac{{6{\rm{*}}{{\left( 3 \right)}^2} + 3{\rm{*}}3 - 12}}{{2{\rm{*}}{{\left(
3 \right)}^2} + 3 +1}}$ = $\frac{{51}}{{22}}$
2) Compute the following limit:
a) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$0 $\frac{{4{{\rm{x}}^3} - {{\rm{x}}^2} + 2{\rm{x}}}}{{3{{\rm{x}}^2}
+ 4{\rm{x}}}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0
$\frac{{4{{\rm{x}}^3} - {{\rm{x}}^2} + 2{\rm{x}}}}{{3{{\rm{x}}^2} +
4{\rm{x}}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0
$\frac{{4{{\rm{x}}^3} - {{\rm{x}}^2} + 2{\rm{x}}}}{{3{{\rm{x}}^2} +
4{\rm{x}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0
$\frac{{{\rm{x}}\left( {4{{\rm{x}}^2} - {\rm{x}} + 2} \right)}}{{{\rm{x}}\left(
{3{\rm{x}} + 4} \right)}}$ = $\frac{{4.0 - 0 + 2}}{{3.0 + 4}} = \frac{1}{2}$
b) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$4 $\frac{{{{\rm{x}}^3} - 64}}{{{{\rm{x}}^2} - 16}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$4
$\frac{{{{\rm{x}}^3} - 64}}{{{{\rm{x}}^2} - 16}}$
When x = 4, the given function takes the form $\frac{0}{0}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$4 $\frac{{{{\rm{x}}^3} - 64}}{{{{\rm{x}}^2} - 16}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$4 $\frac{{{{\left(
{\rm{x}} \right)}^3} - {{\left( 4 \right)}^3}}}{{({{\rm{x}}^{)2}} - {{\left( 4
\right)}^2}}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$4$\frac{{\left(
{{\rm{x}} - 4} \right)\left( {{{\rm{x}}^2} + 4{\rm{x}} + 16}
\right)}}{{{\rm{x}} + 4}}$
= $\frac{{{{\left( 4 \right)}^2} + 4{\rm{*}}4 + 16}}{{4 +
4}}$ = 6
c) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{{{\rm{x}}^{\frac{2}{3}}} -
{{\rm{a}}^{\frac{2}{3}}}}}{{{\rm{x}} - {\rm{a}}}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a
$\frac{{{{\rm{x}}^{\frac{2}{3}}} - {{\rm{a}}^{\frac{2}{3}}}}}{{{\rm{x}} -
{\rm{a}}}}$
When x = a, the given function takes the form $\frac{0}{0}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a$\frac{{{{\rm{x}}^{\frac{2}{3}}}
- {{\rm{a}}^{\frac{2}{3}}}}}{{{\rm{x}} - {\rm{a}}}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a$\frac{{{{\left(
{{{\rm{x}}^{\frac{1}{3}}}} \right)}^2} - {{\left( {{{\rm{a}}^{\frac{1}{3}}}} \right)}^2}}}{{{{\left(
{{{\rm{x}}^{\frac{1}{3}}}} \right)}^3} - {{\left( {{{\rm{a}}^{\frac{1}{3}}}}
\right)}^3}}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\left( {{{\rm{x}}^{\frac{1}{3}}} -
{{\rm{a}}^{\frac{1}{3}}}} \right)\left( {{{\rm{x}}^{\frac{1}{3}}} +
{{\rm{a}}^{\frac{1}{3}}}} \right)}}{{\left( {{{\rm{x}}^{\frac{1}{3}}} -
{{\rm{a}}^{\frac{1}{3}}}} \right)\left( {{{\rm{x}}^{\frac{2}{3}}} +
{{\rm{x}}^{\frac{1}{3}}}.{{\rm{a}}^{\frac{1}{3}}} + {{\rm{a}}^{\frac{2}{3}}}}
\right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a$\frac{{{{\rm{x}}^{\frac{1}{3}}} +
{{\rm{a}}^{\frac{1}{3}}}}}{{{{\rm{x}}^{\frac{2}{3}}} +
{{\rm{x}}^{\frac{1}{3}}}.{{\rm{a}}^{\frac{1}{3}}} + {{\rm{a}}^{\frac{2}{3}}}}}$
= $\frac{{{{\rm{a}}^{\frac{1}{3}}} +
{{\rm{a}}^{\frac{1}{3}}}}}{{{{\rm{a}}^{\frac{2}{3}}} + {{\rm{a}}^{\frac{2}{3}}}
+ {{\rm{a}}^{\frac{2}{3}}}}}$
=
$\frac{{2{{\rm{a}}^{\frac{1}{3}}}}}{{3{{\rm{a}}^{\frac{2}{3}}}}}$
= $\frac{2}{{3{{\rm{a}}^{\frac{1}{3}}}}}$
d) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{{{\rm{x}}^2} + 3{\rm{x}} - 4}}{{{\rm{x}} - 1}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1
$\frac{{{{\rm{x}}^2} + 3{\rm{x}} - 4}}{{{\rm{x}} - 1}}$
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{{{\rm{x}}^2} + 3{\rm{x}} - 4}}{{{\rm{x}} - 1}}$
When x = 1, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{{{\rm{x}}^2} + 3{\rm{x}} - 4}}{{{\rm{x}} - 1}}$ =
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1 $\frac{{\left(
{{\rm{x}} + 4} \right)\left( {{\rm{x}} - 1} \right)}}{{{\rm{x}} - 1}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 (x + 4) = 1 + 4 = 5
e) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{{{\rm{x}}^2} - 5{\rm{x}} + 6}}{{{{\rm{x}}^2} -
{\rm{x}} - 2}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2
$\frac{{{{\rm{x}}^2} - 5{\rm{x}} + 6}}{{{{\rm{x}}^2} - {\rm{x}} - 2}}$
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{{{\rm{x}}^2} - 5{\rm{x}} + 6}}{{{{\rm{x}}^2} -
{\rm{x}} - 2}}$
When x = 2, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{\left( {{\rm{x}} - 3)({\rm{x}} - 2}
\right)}}{{\left( {{\rm{x}} + 1} \right)\left( {{\rm{x}} - 2} \right)}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2 $\frac{{{\rm{x}} -
3}}{{{\rm{x}} + 1}}$ = $\frac{{2 - 3}}{{2 + 1}}$ = $ - \frac{1}{3}$
f) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{{{\rm{x}}^2} - 4{\rm{x}} + 4}}{{{{\rm{x}}^2} -
7{\rm{x}} + 10}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2
$\frac{{{{\rm{x}}^2} - 4{\rm{x}} + 4}}{{{{\rm{x}}^2} - 7{\rm{x}} + 10}}$
When x = 2, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2$\frac{{{{\rm{x}}^2} - 4{\rm{x}} + 4}}{{{{\rm{x}}^2} -
7{\rm{x}} + 10}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2
$\frac{{{{\left( {{\rm{x}} - 2} \right)}^2}}}{{2\left( {{\rm{x}} - 2}
\right)\left( {{\rm{x}} - 5} \right)}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{{\rm{x}} - 2}}{{{\rm{x}} - 5}}$ = $\frac{{2 -
2}}{{2 - 5}}$ = 0
g) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {3{\rm{x}}} - \sqrt {2{\rm{x}} +
{\rm{a}}} }}{{2\left( {{\rm{x}} - {\rm{a}}} \right)}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a
$\frac{{\sqrt {3{\rm{x}}} - \sqrt {2{\rm{x}} + {\rm{a}}} }}{{2\left(
{{\rm{x}} - {\rm{a}}} \right)}}$
When x = a, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {3{\rm{x}}} - \sqrt {2{\rm{x}} +
{\rm{a}}} }}{{2\left( {{\rm{x}} - {\rm{a}}} \right)}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a $\frac{{\sqrt {3{\rm{x}}}
- \sqrt {2{\rm{x}} + {\rm{a}}} }}{{2\left( {{\rm{x}} - {\rm{a}}} \right)}}$ *
$\frac{{\sqrt {3{\rm{x}}} + \sqrt {2{\rm{x}} + {\rm{a}}} }}{{\sqrt
{3{\rm{x}}} + \sqrt {2{\rm{x}} + {\rm{a}}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a
$\frac{{3{\rm{x}} - 2{\rm{x}} - {\rm{a}}}}{{2\left( {{\rm{x}} - {\rm{a}}}
\right)\left( {\sqrt {3{\rm{x}}} + \sqrt {2{\rm{x}} + {\rm{a}}} }
\right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{{\rm{x}} - {\rm{a}}}}{{2\left( {{\rm{x}} -
{\rm{a}}} \right)\left( {\sqrt {3{\rm{x}}} + \sqrt {2{\rm{x}} + {\rm{a}}}
} \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{1}{{2\left( {\sqrt {3{\rm{x}}} + \sqrt
{2{\rm{x}} + {\rm{a}}} } \right)}}$ = $\frac{1}{{2\left( {\sqrt
{3{\rm{a}}} + \sqrt {3{\rm{a}}} } \right)}}$ = $\frac{1}{{4\sqrt 3
{\rm{a}}}}$
h) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {2{\rm{x}}} - \sqrt {3{\rm{x}} -
{\rm{a}}} }}{{\sqrt {\rm{x}} - \sqrt {\rm{a}} }}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a
$\frac{{\sqrt {2{\rm{x}}} - \sqrt {3{\rm{x}} - {\rm{a}}} }}{{\sqrt
{\rm{x}} - \sqrt {\rm{a}} }}$
When x = a, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {2{\rm{x}}} + \sqrt {3{\rm{x}} -
{\rm{a}}} }}{{\sqrt {\rm{x}} - \sqrt {\rm{a}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {2{\rm{x}}} - \sqrt {3{\rm{x}} -
{\rm{a}}} }}{{\sqrt {\rm{x}} - \sqrt {\rm{a}} }}$ * $\frac{{\sqrt
{2{\rm{x}}} + \sqrt {3{\rm{x}} + {\rm{a}}} }}{{\sqrt {2{\rm{x}}} +
\sqrt {3{\rm{x}} - {\rm{a}}} }}$ * $\frac{{\sqrt {\rm{x}} + \sqrt
{\rm{a}} }}{{\sqrt {\rm{x}} + \sqrt {\rm{a}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{{{\left( {\rm{x}} \right)}^2} - {{\left( {\sqrt
{3{\rm{x}} - {\rm{a}}} } \right)}^2}}}{{{{\left( {\sqrt {\rm{x}} } \right)}^2}
- \left( {{{\sqrt {\rm{a}} }^2}} \right)}}$. $\frac{{\sqrt {\rm{x}} +
\sqrt {\rm{a}} }}{{\sqrt {2{\rm{x}}} + \sqrt {3{\rm{x}} - {\rm{a}}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\left( {2{\rm{x}} - 3{\rm{x}} + {\rm{a}}}
\right)\left( {\sqrt {\rm{x}} + \sqrt {\rm{a}} } \right)}}{{\left(
{{\rm{x}} - {\rm{a}}} \right)\left( {\sqrt {2{\rm{x}}} + \sqrt
{3{\rm{x}}} - {\rm{a}}} \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{ - \left( {{\rm{x}} - {\rm{a}}} \right)\left(
{\sqrt {\rm{x}} + \sqrt {\rm{a}} } \right)}}{{\left( {{\rm{x}} -
{\rm{a}}} \right)\left( {\sqrt {2{\rm{x}}} + \sqrt {3{\rm{x}}} -
{\rm{a}}} \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a
$\frac{{ - \left( {\sqrt {\rm{x}} + \sqrt {\rm{a}} } \right)}}{{\sqrt
{2{\rm{x}}} + \sqrt {3{\rm{x}}} - {\rm{a}}}}$ = $\frac{{\sqrt
{\rm{a}} + \sqrt {\rm{a}} }}{{\sqrt {2{\rm{a}}} + \sqrt {2{\rm{a}}}
}}$ = $ - \frac{1}{{\sqrt 2 }}$
i) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{\sqrt {2{\rm{x}}} - \sqrt {3 -
{{\rm{x}}^2}} }}{{{\rm{x}} - 1}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1
$\frac{{\sqrt {2{\rm{x}}} - \sqrt {3 - {{\rm{x}}^2}} }}{{{\rm{x}} - 1}}$
When x = 1, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1
$\frac{{\sqrt {2{\rm{x}}} - \sqrt {3 - {{\rm{x}}^2}} }}{{{\rm{x}} - 1}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{\sqrt {2{\rm{x}}} - \sqrt {3 -
{{\rm{x}}^2}} }}{{{\rm{x}} - 1}}$ * $\frac{{\sqrt {2{\rm{x}}} + \sqrt {3
- {{\rm{x}}^2}} }}{{\sqrt {2{\rm{x}}} + \sqrt {3 - {{\rm{x}}^2}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{2{\rm{x}} - 3 + {{\rm{x}}^2}}}{{\left( {{\rm{x}}
- 1} \right)\left( {\sqrt {2{\rm{x}}} + \sqrt {3 - {{\rm{x}}^2}} }
\right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{{{\rm{x}}^2} + 2{\rm{x}} - 3}}{{\left( {{\rm{x}}
- 1} \right)\left( {\sqrt {2{\rm{x}}} + \sqrt {3 - {{\rm{x}}^2}} }
\right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} -
1} \right)}}{{\left( {{\rm{x}} - 1} \right)\left( {\sqrt {2{\rm{x}}} +
\sqrt {3 - {{\rm{x}}^2}} } \right)}}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{{\rm{x}} + 3}}{{\sqrt {2{\rm{x}}} + \sqrt
{3 - {{\rm{x}}^2}} }}$ = $\frac{{1 + 3}}{{\sqrt 2 + \sqrt 2 }}$ =
$\frac{4}{{2\sqrt 2 }}$ = $\sqrt 2 $
j) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{\sqrt {\rm{x}} - \sqrt {6 - {{\rm{x}}^2}}
}}{{{\rm{x}} - 2}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2
$\frac{{\sqrt {\rm{x}} - \sqrt {6 - {{\rm{x}}^2}} }}{{{\rm{x}} - 2}}$
When x = 2, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{\sqrt {\rm{x}} - \sqrt {6 - {{\rm{x}}^2}}
}}{{{\rm{x}} - 2}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{\sqrt {\rm{x}} - \sqrt {6 - {{\rm{x}}^2}}
}}{{{\rm{x}} - 2}}$ * $\frac{{\sqrt {\rm{x}} + \sqrt {6 - {{\rm{x}}^2}}
}}{{\sqrt {\rm{x}} + {\rm{\: }}\sqrt {6 - {{\rm{x}}^2}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{{\rm{x}} - 6 + {{\rm{x}}^2}}}{{\left( {{\rm{x}} -
2} \right)\left( {\sqrt {\rm{x}} + \sqrt {6 - {{\rm{x}}^2}} } \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} -
2} \right)}}{{\left( {{\rm{x}} - 2} \right)\left( {\sqrt {\rm{x}} + \sqrt
{6 - {{\rm{x}}^2}} } \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{{\rm{x}} + 3}}{{\sqrt {\rm{x}} + \sqrt {6 -
{{\rm{x}}^2}} }}$ = $\frac{5}{{\sqrt 2 + \sqrt 2 }}$ = $\frac{5}{{2\sqrt
2 }}$
k) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$64 $\frac{{\sqrt[6]{{\rm{x}}} - 2}}{{\sqrt[3]{{\rm{x}}} -
4}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$64
$\frac{{\sqrt[6]{{\rm{x}}} - 2}}{{\sqrt[3]{{\rm{x}}} - 4}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$64
$\frac{{{{\rm{x}}^{\frac{1}{6}}} - 2}}{{{{\rm{x}}^{\frac{1}{3}}} - 4}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$64
$\frac{{{{\rm{x}}^{\frac{1}{2}}} - 2}}{{{{\left( {{{\rm{x}}^{\frac{1}{6}}}}
\right)}^2} - {{\left( 2 \right)}^2}}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$64 $\frac{{{{\rm{x}}^{\frac{1}{6}}} - 2}}{{\left(
{{{\rm{x}}^{\frac{1}{6}}} + 2} \right)\left( {{{\rm{x}}^{\frac{1}{6}}} - 2}
\right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$64
$\frac{1}{{{{\rm{x}}^{\frac{1}{6}}} + 2}}$ = $\frac{1}{{{{\left( {64}
\right)}^{\frac{1}{6}}} + 2}}$ = $\frac{1}{{2 + 2}}$ = $\frac{1}{4}$
l) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {3{\rm{a}} - {\rm{x}}} - \sqrt
{{\rm{x}} + {\rm{a}}} }}{{4\left( {{\rm{x}} - {\rm{a}}} \right)}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a
$\frac{{\sqrt {3{\rm{a}} - {\rm{x}}} - \sqrt {{\rm{x}} + {\rm{a}}}
}}{{4\left( {{\rm{x}} - {\rm{a}}} \right)}}$
When x = a, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {3{\rm{a}} - {\rm{x}}} - \sqrt
{{\rm{x}} + {\rm{a}}} }}{{4\left( {{\rm{x}} - {\rm{a}}} \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{\sqrt {3{\rm{a}} - {\rm{x}}} - \sqrt
{{\rm{x}} + {\rm{a}}} }}{{4\left( {{\rm{x}} - {\rm{a}}} \right)}}$ *
$\frac{{\sqrt {3{\rm{a}} - {\rm{x}}} + \sqrt {{\rm{x}} + {\rm{a}}}
}}{{\sqrt {3{\rm{a}} - {\rm{x}}} + \sqrt {{\rm{x}} + {\rm{a}}} }}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $\frac{{3{\rm{a}} - {\rm{x}} - {\rm{x}} -
{\rm{a}}}}{{4\left( {{\rm{x}} - {\rm{a}}} \right)\left( {\sqrt {3{\rm{a}} -
{\rm{x}}} } \right) + \sqrt {{\rm{x}} + {\rm{a}})} }}{\rm{\: \: }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a
$\frac{{2{\rm{x}} - 2{\rm{x}}}}{{4\left( {{\rm{x}} - {\rm{a}}} \right)\left(
{\sqrt {3{\rm{a}} - {\rm{x}}} + \sqrt {{\rm{x}} + {\rm{a}}} } \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a$\frac{{ - 2\left( {{\rm{x}} - {\rm{a}}}
\right)}}{{4\left( {{\rm{x}} - {\rm{a}}} \right)\left( {\sqrt {3{\rm{a}} -
{\rm{x}}} + \sqrt {{\rm{x}} + {\rm{a}}} } \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$a $ - \frac{1}{{2\left( {\sqrt {3{\rm{a}} - {\rm{x}}}
+ \sqrt {{\rm{x}} + {\rm{a}}} } \right)}}$ = $\frac{1}{{2\left( {\sqrt
{2{\rm{a}}} + \sqrt {2{\rm{a}}} } \right)}}$ = $ - \frac{1}{{4\sqrt
{2{\rm{a}}} }}$
3) Compute the following limit:
a) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{2{{\rm{x}}^2}}}{{3{{\rm{x}}^2} + 2}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\frac{{2{{\rm{x}}^2}}}{{3{{\rm{x}}^2} + 2}}$
When x = ∞, the given function takes the form
$\frac{{\rm{\infty }}}{{\rm{\infty }}}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{2{{\rm{x}}^2}}}{{3{{\rm{x}}^2} + 2}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞ $\frac{2}{{3 +
\frac{2}{{{{\rm{x}}^2}}}}}$ (Dividing numerator and denominator by x2)
= $\frac{2}{{3 + 0}}$ = $\frac{2}{3}$
b) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{3{{\rm{x}}^2} - 4}}{{4{{\rm{x}}^2}}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\frac{{3{{\rm{x}}^2} - 4}}{{4{{\rm{x}}^2}}}$
When x = ∞, the given function takes the form
$\frac{{\rm{\infty }}}{{\rm{\infty }}}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\frac{{3{{\rm{x}}^2} - 4}}{{4{{\rm{x}}^2}}}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞ $\left( {\frac{3}{4}
- \frac{1}{{{{\rm{x}}^2}}}} \right)$.
= $\frac{3}{4} - 0$ = $\frac{3}{4}$
c) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{4{{\rm{x}}^2} + 3{\rm{x}} + 2}}{{5{{\rm{x}}^2} +
4{\rm{x}} - 3}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\frac{{4{{\rm{x}}^2} + 3{\rm{x}} + 2}}{{5{{\rm{x}}^2} + 4{\rm{x}} - 3}}$
When x = ∞, the given function takes the form
$\frac{{\rm{\infty }}}{{\rm{\infty }}}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{4{{\rm{x}}^2} + 3{\rm{x}} + 2}}{{5{{\rm{x}}^2} +
4{\rm{x}} - 3}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\frac{{4 + \frac{3}{{\rm{x}}} + \frac{2}{{{{\rm{x}}^2}}}}}{{5 +
\frac{4}{{\rm{x}}} - \frac{3}{{{{\rm{x}}^2}}}}}$ (Dividing numerator and
denominator by x2).
= $\frac{{4 + 0 + 0}}{{5 + 0 - 0}}$ = $\frac{4}{5}$
d) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{5{{\rm{x}}^2} + 2{\rm{x}} - 7}}{{3{{\rm{x}}^2} + 5{\rm{x}}
+ 2}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\frac{{5{{\rm{x}}^2} + 2{\rm{x}} - 7}}{{3{{\rm{x}}^2} + 5{\rm{x}} + 2}}$
When x = ∞, the given function takes the form
$\frac{{\rm{\infty }}}{{\rm{\infty }}}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{5{{\rm{x}}^2} + 2{\rm{x}} - 7}}{{3{{\rm{x}}^2} +
5{\rm{x}} + 2}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\frac{{4 + \frac{3}{{\rm{x}}} + \frac{2}{{{{\rm{x}}^2}}}}}{{5 +
\frac{4}{{\rm{x}}} - \frac{3}{{{{\rm{x}}^2}}}}}$ (Dividing numerator and
denominator by x2).
= $\frac{{5 + 0 - 0}}{{3 + 0 + 0}}$ = $\frac{5}{3}$
4) Calculate the following limits:
a) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} - 3} }
\right)$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} - 3} } \right)$
When x = ∞, the given function takes the form ∞ - ∞.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} - 3} }
\right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞ $\left(
{\sqrt {\rm{x}} - \sqrt {{\rm{x}} - 3} } \right)$ * $\frac{{\sqrt
{\rm{x}} + \sqrt {{\rm{x}} - 3} }}{{\sqrt {\rm{x}} + \sqrt
{{\rm{x}} - 3} }}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{{\rm{x}} - {\rm{x}} + 3}}{{\sqrt {\rm{x}} +
\sqrt {{\rm{x}} - 3} }}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{3}{{\sqrt {\rm{x}} + \sqrt {{\rm{x}} - 3}
}}$ = $\frac{3}{{{\rm{\infty }} + {\rm{\infty }}}}$ = 0
b) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {{\rm{x}} - {\rm{a}}} - \sqrt
{{\rm{x}} - {\rm{b}}} } \right)$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\left( {\sqrt {{\rm{x}} - {\rm{a}}} - \sqrt {{\rm{x}} - {\rm{b}}} }
\right)$
When x = ∞, the given function takes the form ∞ - ∞.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {{\rm{x}} - {\rm{a}}} - \sqrt
{{\rm{x}} - {\rm{b}}} } \right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} - 3} }
\right)$ * $\frac{{\sqrt {{\rm{x}} - {\rm{a}}} + \sqrt {{\rm{x}} -
{\rm{b}}} }}{{\sqrt {{\rm{x}} - {\rm{a}}} + \sqrt {{\rm{x}} - {\rm{b}}}
}}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{{\rm{x}} - {\rm{a}} - {\rm{x}} +
{\rm{b}}}}{{\sqrt {{\rm{x}} - {\rm{a}}} + \sqrt {{\rm{x}} - {\rm{b}}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞ $\frac{{{\rm{b}}
- {\rm{a}}}}{{\sqrt {{\rm{x}} - {\rm{a}}} + \sqrt {{\rm{x}} - {\rm{b}}}
}}$ = $\frac{{{\rm{b}} - {\rm{a}}}}{{\sqrt {{\rm{x}} - {\rm{a}}} + \sqrt
{{\rm{x}} - {\rm{b}}} }}$ = $\frac{{{\rm{b}} - {\rm{a}}}}{{{\rm{\infty }} +
{\rm{\infty }}}}$ = 0
c) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {3{\rm{x}}} - \sqrt {{\rm{x}} - 5} }
\right)$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\left( {\sqrt {3{\rm{x}}} - \sqrt {{\rm{x}} - 5} } \right)$
When x = ∞, the given function takes the form ∞ - ∞.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {3{\rm{x}}} - \sqrt {{\rm{x}} - 5} }
\right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞ $\left(
{\sqrt {3{\rm{x}}} - \sqrt {{\rm{x}} - 5} } \right)$ * $\frac{{\sqrt
{3{\rm{x}}} + \sqrt {{\rm{x}} - 5} }}{{\sqrt {3{\rm{x}}} + \sqrt
{{\rm{x}} - 5} }}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{3{\rm{x}} - {\rm{x}} + 5}}{{\sqrt
{3{\rm{x}}} + \sqrt {{\rm{x}} - 5} }}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{2{\rm{x}} + 5}}{{\sqrt {3{\rm{x}}} + \sqrt
{{\rm{x}} - 5} }}$ (Dividing numerator and denominator by x).
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ = $\frac{{2 + \frac{5}{{\rm{x}}}}}{{\sqrt
{\frac{3}{{\rm{x}}}} + \sqrt {\frac{1}{{\rm{x}}} -
\frac{5}{{{{\rm{x}}^2}}}{\rm{\: }}} }}$ = $\frac{{2 + 0}}{{0 + 0}}$ = ∞
d) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\sqrt {\rm{x}} \left( {\sqrt {\rm{x}} - \sqrt
{{\rm{x}} - {\rm{a}}} } \right)$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\sqrt {\rm{x}} \left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} - {\rm{a}}} }
\right)$
When x = ∞, the given function takes the form ∞ - ∞.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\sqrt {\rm{x}} \left( {\sqrt {\rm{x}} - \sqrt
{{\rm{x}} - {\rm{a}}} } \right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} -
{\rm{a}}} } \right)$ * $\frac{{\sqrt {\rm{x}} + \sqrt {{\rm{x}} -
{\rm{a}}} }}{{\sqrt {\rm{x}} + \sqrt {{\rm{x}} - {\rm{a}}} }}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{\sqrt {\rm{x}} \left( {{\rm{x}} - {\rm{x}} +
{\rm{a}}} \right)}}{{\sqrt {\rm{x}} + \sqrt {{\rm{x}} - {\rm{a}}} }}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞ $\frac{{{\rm{a}}\sqrt
{\rm{x}} }}{{\sqrt {\rm{x}} + \sqrt {{\rm{x}} - {\rm{a}}} }}$
(Dividing numerator and denominator by $\sqrt {\rm{x}} $).
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ = $\frac{{\rm{a}}}{{1 + \sqrt {1 -
\frac{{\rm{a}}}{{\rm{x}}}{\rm{\: }}} }}$ = $\frac{{\rm{a}}}{{1 + 1}}$ =
$\frac{{\rm{a}}}{2}$.
e) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {{\rm{x}} - {\rm{a}}} - \sqrt
{{\rm{bx}}} } \right)$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞
$\left( {\sqrt {{\rm{x}} - {\rm{a}}} - \sqrt {{\rm{bx}}} } \right)$
When x = ∞, the given function takes the form ∞ - ∞.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {{\rm{x}} - {\rm{a}}} - \sqrt
{{\rm{bx}}} } \right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\left( {\sqrt {{\rm{x}} - {\rm{a}}} - \sqrt
{{\rm{bx}}} } \right)$ * $\frac{{\sqrt {{\rm{x}} - {\rm{a}}} + \sqrt
{{\rm{bx}}} }}{{\sqrt {{\rm{x}} - {\rm{a}}} + \sqrt {{\rm{bx}}} }}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ $\frac{{{\rm{x}} - {\rm{a}} - {\rm{bx}}}}{{\sqrt
{{\rm{x}} - {\rm{a}}} + \sqrt {{\rm{bx}}} }}$ = x
$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$∞ $\frac{{\left( {1 -
{\rm{b}}} \right){\rm{x}} - {\rm{a}}}}{{\sqrt {{\rm{x}} - {\rm{a}}} +
\sqrt {{\rm{bx}}} }}$ (Dividing numerator and denominator by x).
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ = $\frac{{\left( {1 - {\rm{b}}} \right) -
\frac{{\rm{a}}}{{\rm{x}}}}}{{\sqrt {\frac{1}{{\rm{x}}} -
\frac{{\rm{a}}}{{{{\rm{x}}^2}}}{\rm{\: }}} + \sqrt
{\frac{{\rm{b}}}{{\rm{x}}}} }}$ = ∞ if b ≠ 1.
If b = 1, then x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$∞ = $\frac{{\left( {1 - {\rm{b}}} \right){\rm{x}} -
{\rm{a}}}}{{\sqrt {{\rm{x}} - {\rm{a\: }}} + \sqrt {{\rm{bx}}} }}$
= $\frac{{ - {\rm{a}}}}{{{\rm{\infty }} + {\rm{\infty }}}}$
= 0
5. a) x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 $\frac{{{\rm{x}} - \sqrt {8 - {{\rm{x}}^2}} }}{{\sqrt
{{{\rm{x}}^2} + 12} - 4}}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2
$\frac{{{\rm{x}} - \sqrt {8 - {{\rm{x}}^2}} }}{{\sqrt {{{\rm{x}}^2} + 12}
- 4}}$
When x = 2, the given function takes the form $\frac{0}{0}$
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2
$\frac{{{\rm{x}} - \sqrt {8 - {{\rm{x}}^2}} }}{{\sqrt {{{\rm{x}}^2} + 12}
- 4}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2
$\frac{{{\rm{x}} - \sqrt {8 - {{\rm{x}}^2}} }}{{\sqrt {{{\rm{x}}^2} + 12}
- 4}}$ * $\frac{{{\rm{x}} + \sqrt {8 - {{\rm{x}}^2}} }}{{{\rm{x}} + \sqrt {8 -
{{\rm{x}}^2}} }}$ * $\frac{{\sqrt {{{\rm{x}}^2} + 12{\rm{\: }}} +
4}}{{\sqrt {{{\rm{x}}^2} + 12} + 4}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2$\frac{{\left( {{{\rm{x}}^2} - 8 + {{\rm{x}}^2}}
\right)\left( {\sqrt {{{\rm{x}}^2} + 12} + 4} \right)}}{{\left(
{{{\rm{x}}^2} + 12 - 16} \right)\left( {{\rm{x}} + \sqrt {8 - {{\rm{x}}^2}} }
\right)}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2$\frac{{2\left( {{{\rm{x}}^2} - 4} \right)\left( {\sqrt
{{{\rm{x}}^2} + 12} + 4} \right)}}{{\left( {{{\rm{x}}^2} - 4}
\right)\left( {{\rm{x}} + \sqrt {8 - {{\rm{x}}^2}} } \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$2 = $\frac{{2\left( {\sqrt {{{\rm{x}}^2} + 12}
+ 4} \right)}}{{\left( {{\rm{x}} + \sqrt {8 - {{\rm{x}}^2}} } \right)}}$=
$\frac{{2\left( {\sqrt {4 + 12} + 4} \right)}}{{2 + \sqrt {8 - 4} }}$ =
$\frac{{2{\rm{*}}8}}{4}$ = 4
b) x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1
$\frac{{{\rm{x}} - \sqrt {2 - {{\rm{x}}^2}} }}{{2{\rm{x}} - \sqrt {2 +
2{{\rm{x}}^2}} }}$
Solution:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1
$\frac{{{\rm{x}} - \sqrt {2 - {{\rm{x}}^2}} }}{{2{\rm{x}} - \sqrt {2 +
2{{\rm{x}}^2}} }}$
When x = 1, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 $\frac{{{\rm{x}} - \sqrt {2 - {{\rm{x}}^2}} }}{{2{\rm{x}}
- \sqrt {2{\rm{x}} + 2{{\rm{x}}^2}} }}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1$\frac{{{\rm{x}} - \sqrt {2 - {{\rm{x}}^2}} }}{{2{\rm{x}}
- \sqrt {2{\rm{x}} + 2{{\rm{x}}^2}} }}$ * $\frac{{{\rm{x}} + \sqrt {2 - {{\rm{x}}^2}}
}}{{{\rm{x}} + \sqrt {2 - {{\rm{x}}^2}} }}$ * $\frac{{2{\rm{x}} + \sqrt {2 +
2{{\rm{x}}^2}} }}{{2{\rm{x}} + \sqrt {2 + 2{{\rm{x}}^2}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1$\frac{{\left( {{{\rm{x}}^2} - 2 + {{\rm{x}}^2}}
\right)\left( {2{\rm{x}} + \sqrt {2 + 2{{\rm{x}}^2}} } \right)}}{{\left(
{4{{\rm{x}}^2} - 2 - 2{{\rm{x}}^2}} \right)\left( {{\rm{x}} + \sqrt {2 -
{{\rm{x}}^2}} } \right)}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1$\frac{{\left( {2{{\rm{x}}^2} - 2} \right)\left(
{2{\rm{x}} + \sqrt {2 + 2{{\rm{x}}^2}} } \right)}}{{\left( {2{{\rm{x}}^2} - 2}
\right)\left( {{\rm{x}} + \sqrt {2 - {{\rm{x}}^2}} } \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$1 = $\frac{{2{\rm{x}} + \sqrt {2 + 2{{\rm{x}}^2}}
}}{{{\rm{x}} + \sqrt {2 - {{\rm{x}}^2}} }}$=$\frac{{2 + 2}}{{1 + 1}}$ = 2