Question

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

• A. $a,b,c$ are in A.P
• B. $a,b,c$ are in G.P
• C. $a,b,c$ are in H.P
• D. none of these

Answer 1

Answer: (D)

none of these

Given, limit is ${lim}_{x\to 0}\frac{x^a{\sin }^{b}x}{\sin \left(x^c\right)}$

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

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

The above limit non-zero valu only when

$a+b-c=0$