Question

If $\underset{x\to 0}{lim}\frac{x^a{\sin }^{b}x}{\sin \left(x^c\right)}$, where $a,b,c\in R\sim \left\{0\right\}$, exists and has and has non-zero value. Then

• A. $a+c=b$
• B. $b+c=a$
• C. $a+b=c$
• D. None of these

Answer 1

The correct option is C $a+b=c$

$\underset{x\to 0}{lim}\frac{x^a{\sin }^{b}x}{\sin x^c}$

$=\underset{x\to 0}{lim}{x}^a{\left(\frac{\sin x}{x}\right)}^b\left(\frac{x^c}{\sin x^c}\right)x^{b-c}$

$=\underset{x\to 0}{lim}{x}^{a+b-c}$..................$\because {lim}_{x\to 0}\left(\frac{\sin x}{x}\right)=1$

This limit will have non-zero value if $a+b=c.$