A) \[\frac{4}{3}{{(in4)}^{2}}\]
B) \[\frac{4}{3}{{(In4)}^{3}}\]
C) \[\frac{3}{2}{{(In4)}^{2}}\]
D) \[\frac{3}{2}{{(In4)}^{3}}\]
Correct Answer: B
Solution :
[b] \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{{{x}^{2}}}{4}\log (1+3x)}\] \[=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{({{4}^{x}}-1)}^{3}}}{{{x}^{3}}}.\frac{{{(x/2)}^{2}}}{\sin {{x}^{2}}/4}.\frac{3x}{\log (1+3x)}.\frac{4}{3}\] \[=\frac{4}{3}{{(lo{{g}_{e}}4)}^{3}}.1.{{\log }_{e}}(e)=\frac{4}{3}{{(lo{{g}_{e}}4)}^{3}}.\]
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