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Q.
$\displaystyle\lim _{x \rightarrow 0} \frac{x^{a} \sin ^{b} x}{\sin \left(x^{c}\right)}$, where $a, b, c \in R \sim\{0\}$, exists and has non-zero value. Then,
Limits and Derivatives
Solution:
$\displaystyle\lim _{x \rightarrow 0} \frac{x^{a} \sin ^{b} x}{\sin x^{c}}$
$=\displaystyle\lim _{x \rightarrow 0} x^{a}\left(\frac{\sin x}{x}\right)^{b}\left(\frac{x^{c}}{\sin x^{c}}\right) x^{b-c}$
$=\displaystyle\lim _{x \rightarrow 0} x^{a+b-c}$
This limit will have non-zero value if $a+b=c$.