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  • The value of p for which the function <img alt="\mathrm{f(x)=\frac{\left(4^x-1\right)^3}{\sin \left(\frac{x}{p}\right) \log \left(1+\frac{x^2}{3}\right)}, x \neq 0=12(\log 4)^3, x=0}" src="htt

The value of p for which the function \mathrm{f(x)=\frac{\left(4^x-1\right)^3}{\sin \left(\frac{x}{p}\right) \log \left(1+\frac{x^2}{3}\right)}, x \neq 0=12(\log 4)^3, x=0} may be continuous at x=0 is 

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

none of these 

 


Answers (1)

best_answer

For continuity \lim = value =12(\log 4)^3 \quad (given)

\mathrm{ \begin{gathered} f(x)=\frac{\left(4^x-1\right)^3}{\sin \left(\frac{x}{p}\right) \log \left\{1+\left(\frac{x^2}{3}\right)\right\}} x \neq 0 \\\\ \lim _{x \rightarrow 0} \frac{\left(\frac{4^x-1}{x}\right)^3 \cdot x^3}{\frac{\sin (x / p)}{(x / p)} \cdot\left(\frac{x}{p}\right) \cdot \frac{x^3}{3} \ln \left\{1+\frac{x^2}{3}\right\}^{3 / x^2}}=3 p(\log 4)^3 \\\\ {\left[? \lim _{x \rightarrow 0}\left\{1+\frac{x^2}{3}\right\}^{3 / x^2}=e \text { and } \lim _{x \rightarrow 0} \frac{\sin (x / p)}{(x / p)}=1\right]} \\\\ \therefore \quad 3 p(\log 4)^3=12(\log 4)^3 \Rightarrow p=4 . \end{gathered} }

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Irshad Anwar

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