question_answer

Determine f(0), so that the function f(x) defined by \[f(x)=\frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{4}\log \left( 1+\frac{{{x}^{2}}}{3} \right)}\] becomes continuous at x = 0.     

Answer:

For \[f(x)\] to be continuous at x = 0, we must have \[\underset{x\to \,\,0}{\mathop{\lim }}\,f(x)=f(0)\] \[\Rightarrow \]   \[f(0)=\underset{x\to 0}{\mathop{\lim }}\,f(x)\] \[=\underset{x\to \,\,0}{\mathop{\lim }}\,f(x)\frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{4}\log \left( 1+\frac{{{x}^{2}}}{3} \right)}\] \[=\frac{{{(lo{{g}_{e}}\,4)}^{3}}}{\frac{1}{4}\times \frac{1}{3}}\] \[=12\,{{(lo{{g}_{e}}\,4)}^{3}}\]