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  • If  <img alt="\lim _{x \rightarrow 0}=\frac{x^{a} \sin ^{b} x}{\sin \lef

If  \lim _{x \rightarrow 0}=\frac{x^{a} \sin ^{b} x}{\sin \left(x^{c}\right)}, where a, b, c \in R-\{0\}, exists and has non zero value, then

Option: 1

a+c=b


Option: 2

b+c=a


Option: 3

a+b=c


Option: 4

none of these


Answers (1)

best_answer

\lim _{x \rightarrow 0} \frac{x^a \sin ^b x}{\sin \left(x^c\right)}=\lim _{x \rightarrow 0} x^a\left(\frac{\sin x}{x}\right)^b \frac{x^c}{\sin \left(x^c\right)} \cdot x^{b-c}

=\lim _{x \rightarrow 0} x^{a+b-c}\left(\frac{\sin x}{x}\right)^b \frac{x^c}{\sin \left(x^2\right)}

This limit will havenon-zero value if  a+b=c

Posted by

vinayak

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