Question

If $m,n\in N,\underset{x\to 0}{\lim }\frac{\sin x^n}{(\sin x{)}^m}$ is

• A. 1 , if $n=m$
• B. 0, if $n>m$
• C. $\infty$, if $n<m$
• D. $n/m$, if $n<m$

Answer 1

Answer: (A), (B), (C)

$L=\underset{x\to 0}{\lim }\frac{\sin x^n}{(\sin x{)}^m}=\underset{x\to 0}{\lim }\frac{\frac{\sin x^n}{x^n}{x}^n}{\frac{(\sin x{)}^m}{x^m}{x}^m}=\underset{x\to 0}{\lim }{x}^{n-m}$
If $n=m$, then
$L=(\text{ a very small value near to zero }{)}^{\text{exactly zero }}=1$
If $n>m$, then
$L=(\text{ a very small value near to zero }{)}^{\text{positive integer }}=0$
If $n<m$, then
$L=(\text{ a very small value near to zero }{)}^{\text{negative integer }}=\infty$