Question

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

The correct options are
A 1, if n $=$ m
B 0, if n > m
C $\infty$, if n < m

$L=\underset{x\to 0}{lim}\frac{sin{x}^n}{(sinx{)}^m}=\underset{x\to 0}{lim}\frac{\frac{sin{x}^n}{x^n}{x}^n}{\frac{(sinx{)}^m}{x^m}{x}^m}=\underset{x\to 0}{lim}{x}^{n-m}$

If $n=m$, then
$L=$ (a very small value near to zero)${}^{exactlyzero}=1$

If $n>m$, then
$L=$ (a very small value near to zero)${}^{positiveinteger}=0$

If $n<m$, then
$L=$ (a very small value near to zero)${}^{negativeinteger}=\infty$