Question

Evaluate the Given limit: $\lim_{x\rightarrow 0}(cosec\,x-cot\,x)$

Answer 1

$\lim_{x\rightarrow 0}(cosec\,x-cot\,x)$
At x = 0, the value of the given function takes the form ∞ to -∞.
Now,
=$\lim_{x\rightarrow 0}(cosec\,x-cot\,x)$
= $\lim_{x\rightarrow 0}(\frac{1}{x}-\frac{cos\,x}{sin\,x})$
= $\lim_{x\rightarrow 0}({1}-\frac{cos\,x}{sin\,x})$
= $\lim_{x\rightarrow 0}$ $\frac{1-\frac{cis\,x}{x}}{\frac{x}{\frac{son\,x}{x}}}$
=$\frac{\lim_{x\rightarrow 0}1-\frac{cos\,x}{x}}{\lim_{x\rightarrow 0}\frac{sin\,x}{x}}$
= $\frac{0}{1}$ [= $\lim_{x\rightarrow 0}$ ( $1-\frac{cos\,x}{x}$) = 0 and $\lim_{x\rightarrow 0}$ $\frac{sin\,x}{x}$ = 1]
=0