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The value of p for which the function

\mathrm{f}(\mathrm{x})=\frac{\left(4^x-1\right)^3}{\sin (x / p) \log \left\{1+\left(x^2 / 3\right)\right\}}, x \neq 0

=12(\log 4)^3, x=0

may be continuous at \mathrm{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( given )

f(x)=\frac{\left(4^x-1\right)^3}{\sin \frac{x}{p} \log \left\{1+\frac{x^2}{3}\right\}} \\

\ln _{x \rightarrow 0} f(x) \ln _{x \rightarrow 0} \frac{\left(4^x-1\right)^3}{\frac{x}{p}\left\{\frac{x^2}{3} \cdots\right\}}=\ln _{x \rightarrow 0} 3 p\left(\frac{4^x-1}{x}\right)^3 \\

                                                                 (L' Hsopital) 

 ={ }_{x \rightarrow 0} 3 p\left(\frac{4^x \log 4}{1}\right)^3 \\                                    

=3 p .(\log 4)^3 \\

\therefore \quad 3 p .(\log 4)^3=12 .(\log 4)^3 

\therefore p=4.

Posted by

Pankaj Sanodiya

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