Question

$\underset{x\to 0}{lim}\frac{\sin x^n}{{\left(\sin x\right)}^m}(m<n)$ is equal to

• A. $1$
• B. $0$
• C. $\frac{n}{m}$
• D. None of he above

Answer 1

Answer: (B)

$0$

${lim}_{x\to 0}\frac{\sin x^n}{{\left(\sin x\right)}^m}\times \frac{x^n}{x^n}\times \frac{x^m}{x^m}$

$={lim}_{x\to 0}\frac{\sin x^n}{x^n}\times {\left(\frac{x}{\sin x}\right)}^m\times \frac{x^n}{x^m}$

$=1\times 1\times {lim}_{x\to 0}{x}^{n-m}$

$=0^{n-m}=0(n>m\Rightarrow n-m>0)$