Question

The value of p for which the function $f\left(x\right)=\left[\begin{array}{cc}\frac{(4^x-1{)}^3}{sin\frac{x}{p}log[1+\frac{x^2}{3}]},& x\ne 0\\ 12(log4{)}^3,& x=0\end{array}\right],$ $x\ne 0$ may be continuous at x = 0, is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is D

4

For f(x) to be continuous at x = 0,

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}{\left(\frac{4^x-1}{x}\right)}^3\times \left(\frac{\frac{x}{p}}{sin\frac{x}{p}}\right).\frac{p{x}^2}{log(1+\frac{x^2}{3})}=(log4{)}^3.1.p.\underset{x\to 0}{lim}\left(\frac{x^2}{\frac{1}{3}{x}^2-\frac{1}{18}{x}^4+\dots }\right)\underset{x\to 0}{lim}f(x)=3p(log4{)}^4\Rightarrow 12(log4{)}^3=3p(log4{)}^{3}p=4$