A) 1
B) 2
C) 3
D) None of these
Correct Answer: D
Solution :
[d] For \[f(x)\] to be continuous at \[x=0\], we should have \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=f(0)=12{{(\log \,\,4)}^{3}}\] \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{{{4}^{x}}-1}{x} \right)}^{3}}\times \frac{\left( \frac{x}{p} \right)}{\left( \sin \frac{x}{p} \right)}\cdot \frac{p{{x}^{2}}}{\log \left( 1+\frac{1}{3}{{x}^{2}} \right)}\] \[={{(log\,4)}^{3}}\cdot 1\cdot p\cdot \underset{x\to 0}{\mathop{\lim }}\,\left( \frac{{{x}^{2}}}{\frac{1}{3}{{x}^{2}}-\frac{1}{18}{{x}^{4}}+...} \right)\] \[=3p{{(log\,\,4)}^{3}}\cdot \text{Hence}\,\,p=4\]
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