Exercise 16.2
1.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinax}}}}{{\rm{x}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\ \to\end{array}$0 $\frac{{{\rm{sinax}}}}{{\rm{x}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\ \to\end{array}$0 $\frac{{{\rm{sinax}}}}{{{\rm{ax}}}}.{\rm{a}}$ = 1.a = a.
2.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{tanbx}}}}{{\rm{x}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{tanbx}}}}{{\rm{x}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinbx}}}}{{{\rm{x}}.{\rm{cosbx}}}}.{\rm{a}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinbx}}}}{{{\rm{bx}}}}$.b.$\frac{1}{{{\rm{cosbx}}}}$ = 1.b.1 = b.
3.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinmx}}}}{{{\rm{sinnx}}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinmx}}}}{{{\rm{sinnx}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinmx}}}}{{{\rm{mx}}}}.{\rm{m}}.\frac{1}{{\frac{{{\rm{sinnx}}}}{{{\rm{nx}}}}.{\rm{n}}}}$ = 1.m.$\frac{1}{{1.{\rm{n}}}}$ = $\frac{{\rm{m}}}{{\rm{n}}}$.
4.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{tanax}}}}{{{\rm{tanbx}}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{tanax}}}}{{{\rm{tanbx}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\frac{{{\rm{sinax}}}}{{{\rm{cosax}}}}}}{{\frac{{{\rm{sinbx}}}}{{{\rm{cosbx}}}}}}{\rm{\: }}$= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinax}}}}{{{\rm{cosax}}}}.\frac{{{\rm{cosbx}}}}{{{\rm{sinbx}}}}$
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 = $\frac{{{\rm{sinax}}}}{{{\rm{ax}}}}.{\rm{a}}.\frac{1}{{{\rm{cosax}}}}.\frac{{{\rm{cosbx}}}}{{\frac{{{\rm{sinbx}}}}{{{\rm{bx}}}}.{\rm{b}}}}$ = 1.a.1.$\frac{1}{{1.{\rm{b}}}}$ = $\frac{{\rm{a}}}{{\rm{b}}}$.
5.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinpx}}}}{{\tan {\rm{qx}}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinpx}}}}{{{\rm{tanqx}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinpx}}.{\rm{cosqx}}}}{{{\rm{sinqx}}}}{\rm{\: }}$= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\frac{{{\rm{sinpx}}}}{{{\rm{px}}}}.{\rm{p}}.{\rm{cosqx}}}}{{\frac{{{\rm{sinqx}}}}{{{\rm{qx}}}}.{\rm{q}}}}$ = $\frac{{1.{\rm{p}}.1}}{{1.{\rm{q}}}}$ = $\frac{{\rm{p}}}{{\rm{q}}}$.
6.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a $\frac{{{\rm{sin}}\left( {{\rm{x}} - {\rm{a}}} \right)}}{{{{\rm{x}}^2} - {{\rm{a}}^2}}}$
When x = a, the given function takes the form $\frac{0}{0}$.
Or, x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a $\frac{{\sin \left( {{\rm{x}} - {\rm{a}}} \right)}}{{{{\rm{x}}^2} - {{\rm{a}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a $\frac{{\sin \left( {{\rm{x}} - {\rm{a}}} \right)}}{{\left( {{\rm{x}} + {\rm{a}}} \right)\left( {{\rm{x}} - {\rm{a}}} \right)}}{\rm{\: }}$= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$a $\frac{{\sin \left( {{\rm{x}} - {\rm{a}}} \right)}}{{{\rm{x}} - {\rm{a}}}}.\frac{1}{{{\rm{x}} + {\rm{a}}}}$ = $\left\{ {{\rm{\: }}\left( {{\rm{x}} - {\rm{a}}} \right)\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}0\frac{{\sin \left( {{\rm{x}} - {\rm{a}}} \right)}}{{{\rm{x}} - {\rm{a}}}}} \right\}\left\{ {{\rm{\: x\: }}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}{\rm{a}}\frac{1}{{{\rm{x}} + {\rm{a}}}}} \right\}$
= 1. $\frac{1}{{{\rm{a}}.{\rm{a}}}}$ = $\frac{1}{{2{\rm{a}}}}$.
7.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$p $\frac{{{{\rm{x}}^2} - {{\rm{p}}^2}}}{{{\rm{tan}}\left( {{\rm{x}} - {\rm{p}}} \right)}}$
When x = p, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$p $\frac{{{{\rm{x}}^2} - {{\rm{p}}^2}}}{{{\rm{tan}}\left( {{\rm{x}} - {\rm{p}}} \right)}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$p $\frac{{{{\rm{x}}^2} - {{\rm{p}}^2}}}{{\frac{{\sin \left( {{\rm{x}} - {\rm{p}}} \right)}}{{\cos \left( {{\rm{x}} - {\rm{p}}} \right)}}}}{\rm{\: }}$= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$p $\frac{{\left( {{\rm{x}} + {\rm{p}}} \right)\left( {{\rm{x}} - {\rm{p}}} \right).{\rm{cos}}\left( {{\rm{x}} - {\rm{p}}} \right)}}{{{\rm{sin}}\left( {{\rm{x}} - {\rm{p}}} \right)}}$
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$p $\frac{{{\rm{x}} - {\rm{p}}}}{{\sin \left( {{\rm{x}} - {\rm{p}}} \right)}}.\left( {{\rm{x}} + {\rm{p}}} \right).{\rm{cos}}\left( {{\rm{x}} - {\rm{p}}} \right)$
= (x – p) $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{1}{{\frac{{\left( {\sin \left( {{\rm{x}} - {\rm{p}}} \right)} \right)}}{{{\rm{x}} - {\rm{p}}}}}}$. x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$p $\left( {{\rm{x}} + {\rm{p}}} \right).{\rm{cos}}\left( {{\rm{x}} - {\rm{p}}} \right)$= 1.(p + p).1 = 2p.
8.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinax}}.{\rm{cosbx}}}}{{{\rm{sincx}}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinax}}.{\rm{cosbx}}}}{{{\rm{sincx}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\frac{{{\rm{sinax}}}}{{{\rm{ax}}}}.{\rm{a}}.{\rm{acosbx}}}}{{\frac{{{\rm{sincx}}}}{{{\rm{cx}}}}.{\rm{c}}}}$ = $\frac{{1.{\rm{a}}.1}}{{1.{\rm{c}}}}$ = $\frac{{\rm{a}}}{{\rm{c}}}$.
9.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{1 - {\rm{cosbx}}}}{{{{\rm{x}}^2}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{1 - {\rm{cosbx}}}}{{{{\rm{x}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{2{{\sin }^2}\frac{{\rm{x}}}{2}}}{{{{\rm{x}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0$\left[ {2.{{\left( {\frac{{\sin \frac{{\rm{x}}}{2}}}{{\frac{{\rm{x}}}{2}}}} \right)}^2}.\frac{1}{4}} \right]$.
= $\frac{1}{2}{\left( 1 \right)^2}$ = $\frac{1}{2}$.
10.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{1 - {\rm{cos}}6{\rm{x}}}}{{{{\rm{x}}^2}}}$
When x = 0 , the given function takes the form $\frac{0}{0}$.
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{1 - {\rm{cos}}6{\rm{x}}}}{{{{\rm{x}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{2{{\sin }^2}3{\rm{x}}}}{{{{\rm{x}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0$\left[ {2.{{\left( {\frac{{\sin 3{\rm{x}}}}{{3{\rm{x}}}}} \right)}^2}.9} \right]$ = 18.
11.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{1 - {\rm{cos}}9{\rm{x}}}}{{{{\rm{x}}^2}}}$
When x = 0, the given function takes the form $\frac{0}{0}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{1 - {\rm{cos}}9{\rm{x}}}}{{{{\rm{x}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{2{{\sin }^2}\frac{{9{\rm{x}}}}{2}}}{{{{\rm{x}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{2{{\left( {\sin \frac{{9{\rm{x}}}}{2}} \right)}^2}}}{{{{\rm{x}}^2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\left[ {2{{\left( {\frac{{\sin \frac{{9{\rm{x}}}}{2}}}{{\frac{{9{\rm{x}}}}{2}}}} \right)}^2}\left( {\frac{{81}}{4}} \right)} \right]$ = $\frac{{81}}{2}$.
12.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{cosax}} - {\rm{cosbx}}}}{{{{\rm{x}}^2}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= $\frac{{2\sin \frac{{{\rm{ax}} + {\rm{bx}}}}{2}.\sin \frac{{{\rm{bx}} - {\rm{ax}}}}{2}}}{{{{\rm{x}}^2}}}$ = $\frac{{\frac{{2\sin \left( {{\rm{a}} + {\rm{b}}} \right){\rm{x}}}}{2}}}{{\rm{x}}}.\frac{{\frac{{\sin \left( {{\rm{b}} - {\rm{a}}} \right){\rm{x}}}}{2}}}{{\rm{x}}}$
= $\frac{{2\sin \frac{{\left( {{\rm{a}} + {\rm{b}}} \right){\rm{x}}}}{2}}}{{\left( {\frac{{{\rm{a}} + {\rm{b}}}}{2}} \right){\rm{x}}.\frac{2}{{{\rm{a}} + {\rm{b}}}}}}{\rm{\: }}.\frac{{\sin \frac{{\left( {{\rm{b}} - {\rm{a}}} \right){\rm{x}}}}{2}}}{{\left( {\frac{{{\rm{b}} - {\rm{a}}}}{2}} \right){\rm{x}}.\frac{2}{{{\rm{b}} - {\rm{a}}}}}}$ = $2.\frac{{{\rm{a}} + {\rm{b}}}}{2}.\frac{{{\rm{b}} - {\rm{a}}}}{2}$ = $\frac{{{{\rm{b}}^2} - {{\rm{a}}^2}}}{2}$.
13.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sinax}} - {\rm{sinbx}}}}{{\rm{x}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\left( {\frac{{{\rm{sinax}}}}{{\rm{x}}} - \frac{{{\rm{sinbx}}}}{{\rm{x}}}} \right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\left( {\frac{{{\rm{sinax}}}}{{{\rm{ax}}}}{\rm{\: \: }}.{\rm{a}} - \frac{{{\rm{sinbx}}}}{{{\rm{bx}}}}.{\rm{b}}} \right){\rm{\: }}$
= 1.a – 1.b = a – b.
14.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{1 - {\rm{cospx}}}}{{1 - {\rm{cosqx}}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{2{{\sin }^2}\frac{{{\rm{px}}}}{2}}}{{{{\sin }^2}\frac{{{\rm{qx}}}}{2}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\left( {\frac{{{\rm{sinpx}}}}{2}} \right)}^2}}}{{{{\left( {\frac{{{\rm{sinqx}}}}{2}} \right)}^2}}}{\rm{\: }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\left[ {{{\left( {\frac{{\sin \frac{{{\rm{px}}}}{2}}}{{\frac{{{\rm{px}}}}{2}}}} \right)}^2}.\frac{{{{\rm{p}}^2}}}{4}.\frac{1}{{\left( {\frac{{\frac{{{\rm{sinqx}}}}{2}}}{{\frac{{{\rm{qx}}}}{2}}}} \right)\frac{{{{\rm{q}}^2}}}{4}{\rm{\: \: }}}}} \right]$ = $\frac{{{{\rm{p}}^2}}}{{{{\rm{q}}^2}}}$.
15.
Soln:
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{tanx}} - {\rm{sinx}}}}{{{{\rm{x}}^3}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\frac{{{\rm{sinx}}}}{{{\rm{cosx}}}} - {\rm{sinx}}}}{{{{\rm{x}}^3}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sin}}2{\rm{x}} - {\rm{sin}}2{\rm{x}}.{\rm{cos}}2{\rm{x}}}}{{{{\rm{x}}^3}.{\rm{cos}}2{\rm{x}}}}{\rm{\: }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sin}}2{\rm{x}}\left( {1 - {\rm{cos}}2{\rm{x}}} \right)}}{{{{\rm{x}}^3}{\rm{cos}}2{\rm{x}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sin}}2{\rm{x}}.2{{\sin }^2}{\rm{x}}}}{{{{\rm{x}}^3}{\rm{cos}}2{\rm{x}}}}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sin}}2{\rm{x}}}}{{2{\rm{x}}}}$.2.2${\left( {\frac{{{\rm{sinx}}}}{{\rm{x}}}} \right)^2}$.$\frac{1}{{{\rm{cos}}2{\rm{x}}}}$ = 1.4.1 = 4.
17.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 (secx – tanx) [∞ – ∞ form]
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 $\left( {\frac{1}{{{\rm{cosx}}}} - \frac{{{\rm{sinx}}}}{{{\rm{cosx}}}}} \right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 $\left( {\frac{{1 - {\rm{sinx}}}}{{\cos {\rm{x}}}}} \right)$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 $\frac{{1 - {\rm{sinx}}}}{{{\rm{cosx}}}}$ * $\frac{{1 + {\rm{sinx}}}}{{1 + {\rm{sinx}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 $\frac{{1 - {{\sin }^2}{\rm{x}}}}{{2{\rm{cosx}}\left( {1 + {\rm{sinx}}} \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 $\frac{{{\rm{cosx}}}}{{1 + {\rm{sinx}}}}$ = $\frac{{\cos \frac{{\rm{\pi }}}{2}}}{{1 + \sin \frac{{\rm{\pi }}}{2}}}$ = 0.
18.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{{\rm{se}}{{\rm{x}}^2}{\rm{x}} - 2{\rm{\: }}}}{{{\rm{tanx}} - 1}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{1 + {{\tan }^2}{\rm{x}} - 2}}{{{\rm{tanx}} - 1}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{{{\tan }^2}{\rm{x}} - 1}}{{{\rm{tanx}} - 1}}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{\left( {{\rm{tanx}} + 1} \right)\left( {{\rm{tanx}} - 1} \right)}}{{{\rm{tanx}} - 1}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 (tanx + 1)
= tan $\frac{{\rm{\pi }}}{4}$ + 1 = 1 + 1 = 2.
19.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 = $\frac{{2 - {\rm{cose}}{{\rm{c}}^2}{\rm{x}}}}{{1 - {\rm{cotx}}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{2 - 1 - {{\cot }^2}{\rm{x}}}}{{1 - {\rm{cotx}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{1 - {{\cot }^2}{\rm{x}}}}{{1 - {\rm{cotx}}}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{\left( {1 + {\rm{cotx}}} \right)\left( {1 - {\rm{cotx}}} \right)}}{{1 - {\rm{cotx}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 (1 + cotx)
= 1 + cot$\frac{{\rm{\pi }}}{4}$ = 1 + 1 = 2.
20.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{{\rm{tanx}} - {\rm{tany}}}}{{{\rm{x}} - {\rm{y}}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\left( {\frac{{\frac{{{\rm{sinx}}}}{{{\rm{cosx}}}} - \frac{{{\rm{siny}}}}{{{\rm{cosy}}}}}}{{{\rm{x}} - {\rm{y}}}}} \right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{{\rm{sinx}}.{\rm{cosy}} - {\rm{cosx}}.{\rm{siny}}}}{{{\rm{cosx}}.{\rm{cosy}}\left( {{\rm{x}} - {\rm{y}}} \right)}}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{\sin \left( {{\rm{x}} - {\rm{y}}} \right)}}{{{\rm{cosx}}.{\rm{cosy}}\left( {{\rm{x}} - {\rm{y}}} \right)}}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{\sin \left( {{\rm{x}} - {\rm{y}}} \right)}}{{{\rm{x}} - {\rm{y}}}}$. x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{1}{{{\rm{cosx}}.{\rm{cosy}}}}$
= 1.$\frac{1}{{{\rm{cosy}}.{\rm{cosy}}}}$ = sec2y.
21.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{{\rm{sinx}} - {\rm{siny}}}}{{{\rm{x}} - {\rm{y}}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{2\cos \frac{{{\rm{x}} + {\rm{y}}}}{2}.\sin \frac{{{\rm{x}} - {\rm{y}}}}{2}}}{{{\rm{x}} - {\rm{y}}}}$
= 2 x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y cos $\frac{{{\rm{x}} + {\rm{y}}}}{2}$. x – y $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\sin \frac{{{\rm{x}} - {\rm{y}}}}{2}}}{{\frac{{{\rm{x}} - {\rm{y}}}}{2}.2{\rm{\: }}}}$ = 2.cos$\frac{{2{\rm{y}}}}{2}.\frac{1}{2}$ = cosy.
22.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$y $\frac{{{\rm{cosx}} -
{\rm{cosy}}}}{{{\rm{x}} - {\rm{y}}}}$$\left[
{\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{2\sin \frac{{{\rm{x}} + {\rm{y}}}}{2}.\sin \frac{{{\rm{y}} - {\rm{x}}}}{2}}}{{{\rm{x}} - {\rm{y}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y $\frac{{ - 2\sin \frac{{{\rm{x}} + {\rm{y}}}}{2}.\sin \frac{{{\rm{x}} - {\rm{y}}}}{2}}}{{{\rm{x}} - {\rm{y}}}}$
= –2 x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$y sin $\frac{{{\rm{x}} + {\rm{y}}}}{2}$. (x – y) $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\sin \frac{{{\rm{x}} - {\rm{y}}}}{2}}}{{\frac{{{\rm{x}} - {\rm{y}}}}{2}.2}}$
= –2sin $\frac{{2{\rm{y}}}}{2}.\frac{1}{2}$ = –siny.
23.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\:
}}}\\\to\end{array}$θ
$\frac{{{\rm{xcot}}\theta - \theta
{\rm{cotx}}}}{{{\rm{x}} - \theta }}$$\left[
{\frac{0}{0}{\rm{form}}} \right]$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{{\rm{xcot}}\theta - \theta {\rm{cot}}\theta + \theta {\rm{cot}}\theta - \theta {\rm{cotx}}}}{{{\rm{x}} - \theta }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{\left( {{\rm{x}} - \theta } \right){\rm{cot}}\theta + \theta \left( {{\rm{cot}}\theta - {\rm{cotx}}} \right)}}{{{\rm{x}} - \theta }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $[\left( {{\rm{cot}}\theta + \frac{\theta }{{{\rm{x}} - \theta }}\left\{ {\frac{{{\rm{cos}}\theta }}{{{\rm{sin}}\theta }} - \frac{{{\rm{cosx}}}}{{{\rm{sinx}}}}} \right\}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\left[ {{\rm{cot}}\theta + \frac{\theta }{{{\rm{x}} - \theta }}.\frac{{{\rm{sinxcos}}\theta - {\rm{cosx}}.{\rm{sin}}\theta }}{{{\rm{sin}}\theta .{\rm{sinx}}}}} \right]$
= cotθ + x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{\theta }{{\left( {{\rm{x}} - \theta } \right)}}$.$\frac{{\sin \left( {{\rm{x}} - \theta } \right)}}{{{\rm{sin}}\theta .{\rm{sinx}}}}{\rm{\: \: }}$
= cotθ + x – θ $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{sin}}\left( {{\rm{x}} - \theta } \right)}}{{{\rm{x}} - \theta }}$. x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{\theta }{{{\rm{sin}}\theta .{\rm{sinx}}}}$.
= cotθ + 1.$\frac{\theta }{{{\rm{sin}}\theta .{\rm{sin}}\theta }}$ = cotθ + $\frac{\theta }{{{{\sin }^2}\theta }}$.
24.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{{\rm{xcos}}\theta - \theta {\rm{cosx}}}}{{{\rm{x}} - \theta }}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{{\rm{xcos}}\theta - \theta {\rm{cos}}\theta + \theta {\rm{cos}}\theta - \theta {\rm{cosx}}}}{{{\rm{x}} - \theta }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{\left( {{\rm{x}} - \theta } \right){\rm{cos}}\theta + \theta \left( {{\rm{cos}}\theta - {\rm{cosx}}} \right)}}{{{\rm{x}} - \theta }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\left[ {{\rm{cos}}\theta + \frac{\theta }{{{\rm{x}} - \theta }}.2\sin \frac{{\theta + {\rm{x}}}}{2}.\sin \frac{{{\rm{x}} - \theta }}{2}} \right]$
= cosθ + x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ 2θsin $\frac{{\theta + {\rm{x}}}}{2}$x – θ $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\sin \frac{{{\rm{x}} - \theta }}{2}}}{{\frac{{{\rm{x}} - \theta }}{2}.2}}$
= cosθ + 2θ.sin $\frac{{2\theta }}{2}$.1.$\frac{1}{2}$ = cosθ + θ.sinθ.
25.
Soln:
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1 $\frac{{1 + {\rm{cos\pi x}}}}{{{{\tan }^2}{\rm{\pi x}}}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1 $\frac{{1 + {\rm{cos\pi x}}}}{{{{\tan }^2}{\rm{\pi x}}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$1 $\frac{{\left( {1 + {\rm{cos\pi x}}} \right){{\cos }^2}{\rm{\pi x}}}}{{\left( {1 + {\rm{cos\pi x}}} \right)\left( {1 - {\rm{cos\pi x}}} \right)}}$
= $\frac{{{{\cos }^2}{\rm{\pi }}}}{{1 - {\rm{cos\pi }}}}$
= $\frac{{{{\left( { - 1} \right)}^2}}}{{1 - \left( { - 1} \right)}}$ = $\frac{1}{2}$.
26.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{{\rm{xtan}}\theta - \theta {\rm{tanx}}}}{{{\rm{x}} - \theta }}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{{\rm{xtan\: }}\theta - \theta {\rm{tan}}\theta + \theta {\rm{tan}}\theta - \theta {\rm{tanx}}}}{{{\rm{x}} - \theta }}$.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{{\left( {{\rm{x}} - \theta } \right){\rm{tan}}\theta + \theta \left( {{\rm{tan}}\theta - {\rm{tanx}}} \right)}}{{{\rm{x}} - \theta }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\left[ {{\rm{tan}}\theta + \frac{\theta }{{{\rm{x}} - \theta }}\left\{ {{\rm{tan}}\theta - {\rm{tanx}}} \right\}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\left[ {{\rm{tan}}\theta + \frac{\theta }{{{\rm{x}} - \theta }}\left\{ {\frac{{{\rm{sin}}\theta }}{{{\rm{cos}}\theta }} - \frac{{{\rm{sinx}}}}{{{\rm{cosx}}}}} \right\}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\left[ {{\rm{tan}}\theta + \frac{\theta }{{{\rm{x}} - \theta }}\left\{ {\frac{{{\rm{sin}}\theta .{\rm{cosx}} - {\rm{cos}}\theta .{\rm{sinx}}}}{{{\rm{cos}}\theta .{\rm{cosx}}}}} \right\}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\left[ {{\rm{tan}}\theta + \frac{\theta }{{{\rm{x}} - \theta }}.\frac{{\sin \left( {\theta - {\rm{x}}} \right)}}{{{\rm{cos}}\theta .{\rm{cosx}}}}} \right]$
= tanθ – x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$θ $\frac{\theta }{{{\rm{cos}}\theta .{\rm{cosx}}}}$x – θ $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\sin \left( {{\rm{x}} - \theta } \right)}}{{{\rm{x}} - \theta }}$
= tanθ –$\frac{\theta }{{{\rm{cos}}\theta .{\rm{cos}}\theta }}$.1 = tanθ – θsec2θ.
27.
Soln:
= θ $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{{\rm{cos}}\theta - {\rm{sin}}\theta }}{{\theta - \frac{{\rm{\pi }}}{4}}}$$\left[ {\frac{0}{{0{\rm{form}}}}} \right]$
= θ $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{\sqrt 2 \left( {\frac{1}{{\sqrt 2 }}{\rm{cos}}\theta - \frac{1}{{\sqrt 2 }}{\rm{sin}}\theta } \right)}}{{\theta - \frac{{\rm{\pi }}}{4}}}$
= θ $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{\sqrt 2 \left( {\sin \frac{{\rm{\pi }}}{4}{\rm{cos}}\theta - \cos \frac{{\rm{\pi }}}{4}.{\rm{sin}}\theta } \right)}}{{\theta - \frac{{\rm{\pi }}}{4}}}$
= θ $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/4 $\frac{{\sqrt 2 \sin \left( {\frac{{\rm{\pi }}}{4} - \theta } \right)}}{{\theta - \frac{{\rm{\pi }}}{4}}}$
= θ $--\frac{{\rm{\pi }}}{4}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{ - \sqrt 2 \left( {\theta - \frac{{\rm{\pi }}}{4}} \right)}}{{\theta - \frac{{\rm{\pi }}}{4}}}$
= $ - \sqrt 2 $.1 = $ - \sqrt 2 $.
28.
Soln:
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$c $\frac{{\sqrt {\rm{x}} - \sqrt {\rm{c}} }}{{{\rm{sinx}} - {\rm{sinc}}}}$$\left[ {\frac{0}{0}{\rm{form}}} \right]$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$c $\frac{{\sqrt {\rm{x}} - \sqrt {\rm{c}} }}{{2\sin \frac{{{\rm{x}} - {\rm{c}}}}{2}.\cos \frac{{{\rm{x}} + {\rm{c}}}}{2}}}$ * $\frac{{\sqrt {\rm{x}} + \sqrt {\rm{c}} }}{{\sqrt {\rm{x}} + \sqrt {\rm{c}} }}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$c $\frac{{{\rm{x}} - {\rm{c}}}}{{2\sin \frac{{{\rm{x}} - {\rm{c}}}}{2}.\cos \frac{{{\rm{x}} + {\rm{c}}}}{2}\left( {\sqrt {\rm{x}} + \sqrt {\rm{c}} } \right)}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$c $\frac{2}{{2\left( {\sqrt {\rm{x}} + \sqrt {\rm{c}} } \right).\sin \frac{{\frac{{\frac{{{\rm{x}} - {\rm{c}}}}{2}}}{{{\rm{x}} - {\rm{c}}}}}}{2}.\cos \frac{{{\rm{x}} + {\rm{c}}}}{2}}}$
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$c $\frac{1}{{\left( {\sqrt {\rm{x}} + \sqrt {\rm{x}} } \right)\cos \frac{{{\rm{x}} + {\rm{c}}}}{2}}}$ = $\frac{1}{{\left( {\sqrt {\rm{c}} + \sqrt {\rm{c}} } \right).\cos \frac{{{\rm{c}} + {\rm{c}}}}{2}}}$ = $\frac{1}{{2\sqrt {\rm{c}} .{\rm{cosc}}}}$.
= secc. $\frac{1}{{2\sqrt {\rm{c}} }}$ = $\frac{{{\rm{secc}}}}{{2\sqrt {\rm{c}} }}{\rm{\: }}$.
29.
Soln:
a.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{e}}^{6{\rm{x}}}} - 1}}{{\rm{x}}}$$\left( {\frac{0}{0}} \right)$ = $\left( {{\rm{\: x\: }}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}0{\rm{\: }}\frac{{{{\rm{e}}^{6{\rm{x}}}} - 1}}{{\rm{x}}}} \right)$.6 = 1.6 = 6.
b.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{e}}^{2{\rm{x}}}} - 1}}{{{\rm{x}}{{.2}^{{\rm{x}} + 1}}}},$$\left( {\frac{0}{0}} \right)$ = $\left( {{\rm{\: x\: }}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}0{\rm{\: }}\frac{{{{\rm{e}}^{2{\rm{x}}}} - 1}}{{2{\rm{x}}}}.\frac{2}{{{2^{{\rm{x}} + 1}}}}} \right)$ = 1.$\frac{2}{2}$ = 1.
c.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{e}}^{{\rm{ax}}}} - {{\rm{e}}^{{\rm{bx}}}}}}{{\rm{x}}}$$,\left( {\frac{0}{0}} \right)$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{e}}^{{\rm{ax}}}} - 1 - \left( {{{\rm{e}}^{{\rm{bx}}}} - 1} \right)}}{{\rm{x}}}$ = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{e}}^{{\rm{ax}}}} - 1}}{{\rm{x}}}$ – x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{e}}^{{\rm{bx}}}} - 1}}{{\rm{x}}}$ = $\left( {{\rm{x\: }}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}0{\rm{\: }}\frac{{{{\rm{e}}^{{\rm{ax}}}} - 1}}{{{\rm{ax}}}}} \right).{\rm{a}} - \left( {{\rm{x\: }}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}0{\rm{\: }}\frac{{{{\rm{e}}^{{\rm{bx}}}} - 1}}{{{\rm{bx}}}}} \right).{\rm{b}}$ = 1.a – 1.b = a – b.
d.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{a}}^{\rm{x}}} + {{\rm{b}}^{\rm{x}}} - 2}}{{\rm{x}}}$$,\left( {\frac{0}{0}} \right)$ = = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{{\rm{a}}^{\rm{x}}} - 1}}{{\rm{x}}}$$ + {\rm{\: }} = {\rm{\: x\: }}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}0{\rm{\: }}\frac{{{{\rm{b}}^{\rm{x}}} - 1}}{{\rm{x}}}{\rm{\: }}$= loga + logb = log(ab).
30.
Soln:
a.
x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$2 $\frac{{{\rm{x}} - 2}}{{{\rm{log}}\left( {{\rm{x}} - 1} \right)}}$
let x – 2 = y → x = y + 2.
So, x → 2 ⇒ y → 0.
Now, = x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{{\rm{x}} - 2}}{{{\rm{log}}\left( {{\rm{x}} - 1} \right)}}$ = y$\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\rm{y}}}{{{\rm{log}}\left( {{\rm{y}} + 1} \right)}}$ = $\frac{1}{{{\rm{y}}\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}0{\rm{\: }}\frac{{\log \left( {{\rm{y}} + 1} \right)}}{{\rm{y}}}{\rm{\: }}}}$ = $\frac{1}{1}$ = 1.
b.
= x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 $\frac{{{\rm{cosx}}}}{{\log \left( {{\rm{x}} - \frac{{\rm{\pi }}}{2} + 1} \right)}}$, $\left( {\frac{0}{0}} \right)$
Let x – π/2 = y → y ⇒ x = $\frac{{\rm{\pi }}}{2}$ + y
So, x →$\frac{{\rm{\pi }}}{2}$⇒ y → 0
Now,
=x $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$π/2 $\frac{{{\rm{cosx}}}}{{\log \left( {{\rm{x}} - \frac{{\rm{\pi }}}{2} + 1} \right)}}$
= y $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\frac{{\cos \left( {\frac{{\rm{\pi }}}{2} + {\rm{y}}} \right)}}{{\log \left( {{\rm{y}} + 1} \right)}}$ = y $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $--\frac{{{\rm{siny}}}}{{\log \left( {{\rm{y}} + 1} \right)}}$.
= –y $\begin{array}{*{20}{c}}{{\rm{lim\: }}}\\\to\end{array}$0 $\left( {\frac{{\frac{{\frac{{{\rm{siny}}}}{{\rm{y}}}}}{{\log \left( {{\rm{y}} + 1} \right)}}}}{{\rm{y}}}} \right)$ = $ - \frac{1}{1}$ = –1.
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