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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Evaluate the following limits:

$
\lim _{x \rightarrow a}\left[\frac{\sin x-\sin a}{x-a}\right]
$

✓

Answer

$
\lim _{x \rightarrow a} \frac{\sin x-\sin a}{x-a}
$
Put $x=\mathrm{a}+\mathrm{h}$,
$
\therefore \quad x-\mathrm{a}=\mathrm{h}
$
As $x \rightarrow \mathrm{a}, \mathrm{h} \rightarrow 0$
$
\begin{aligned}
\therefore \quad & \lim _{x \rightarrow \mathbf{a}} \frac{\sin x-\sin a}{x-a} \\
& =\lim _{h \rightarrow 0} \frac{\sin (a+h)-\sin a}{h} \\
& =\lim _{\mathbf{h} \rightarrow 0} \frac{2 \cos \left(\frac{a+h+a}{2}\right) \sin \left(\frac{a+h-a}{2}\right)}{h} \\
& =\lim _{h \rightarrow 0} \frac{2 \cos \left(a+\frac{h}{2}\right) \sin \frac{h}{2}}{h}\\
& =\lim _{h \rightarrow 0} \cos \left(a+\frac{h}{2}\right) \cdot \lim _{h \rightarrow 0} \frac{\sin \left(\frac{h}{2}\right)}{\left(\frac{h}{2}\right)} \\
& =\cos (a+0)(1) \\
& =\cos a
\end{aligned}
$

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