Limits Continuity And Differentiability Question 3
Question: The value of p for which the function $ f(x)= \begin{matrix} \frac{{{(4^{x}-1)}^{3}}}{\sin \frac{x}{p}\log [ 1+\frac{x^{2}}{3} ]},x\ne 0 \\ 12{{(log4)}^{3}},x=0 \\ \end{matrix} . $ may be continuous at $ x=0 $ , is
Options:
A) 1
B) 2
C) 3
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
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 }}{{( \frac{4^{x}-1}{x} )}^{3}}\times \frac{( \frac{x}{p} )}{( \sin \frac{x}{p} )}\cdot \frac{px^{2}}{\log ( 1+\frac{1}{3}x^{2} )} $
$ ={{(log4)}^{3}}\cdot 1\cdot p\cdot \underset{x\to 0}{\mathop{\lim }}( \frac{x^{2}}{\frac{1}{3}x^{2}-\frac{1}{18}x^{4}+…} ) $
$ =3p{{(log4)}^{3}}\cdot Hence p=4 $
BETA
