Question

The value of $\underset{x\to 0}{\lim }\left[\frac{a}{x}-cot\frac{x}{a}\right]$ is

Answer 1

Answer: 0

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