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  4. The value of lim limitsx→0 [(a/x) - cot (x/a)] is

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Q. The value of $\lim\limits_{x\to0} \left[\frac{a}{x} - cot \frac{x}{a}\right]$ is

Limits and Derivatives

Solution:

We have limit $= \lim\limits_{x\to0} \left[\frac{a}{x} - \frac{cos \left(x/a\right)}{sin \left(x/ a\right)}\right] $
$ = \lim\limits_{x\to 0} \left[\frac{a\,sin\left(x/a\right)-x\, cos \left(x/a\right)}{x \,sin\left(x/a\right)}\right]$
$ = a \lim\limits_{x\to 0} \left[\frac{a\,sin\left(x/a\right)-x\, cos \left(x/a\right)}{x^2} \right] \times \frac{(x/a)}{sin(x/a)}$
$ = a \lim\limits_{x\to 0} \left[\frac{a\,sin\left(x/a\right)-x\, cos \left(x/a\right)}{x^2}\right]$ $(\frac{0}{0}$ form$)$
$ = a \lim\limits_{x\to 0} [\frac{cos (x/a) - cos (x/a) + (x/a) sin (x/a)}{2x}]$
$ = 0$