Question

Evaluate: $\underset{x\to 0}{lim}(\csc x-\cot x)$

Answer 1

$\underset{x\to 0}{lim}(\csc x-\cot x)=0$.

Solution :

$cscx-cotx$ = $\frac{1}{\sin x}-\frac{\cos x}{\sin x}=\frac{1-\cos x}{sinx}$ ...... (1)

Series expansion of $\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+.....$

Series expansion of $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+.....$

form the above expansion $1-\cos x=\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+.....$

Now substitute the values of $1-\cos x$ and $\sin x$ in $\left(1\right)$

$\underset{x\to 0}{lim}\frac{\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+.....}{x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+.....}=\frac{0}{1}=0$.