Limits Continuity And Differentiability Question 77
Question: A function f is defined as follows $ f(x)=x^{p}\cos ( \frac{1}{x} ),x\ne 0f(0)=0 $ What conditions should be imposed on p so that f may be continuous at x = 0?
Options:
A) p = 0
B) p > 0
C) p < 0
D) No value of p
Show Answer
Answer:
Correct Answer: B
Solution:
Given function is defined as: $ f(x)= \begin{matrix} x^{p}\cos ( \frac{1}{x} ) & x\ne 0 \\ 0, & x=0 \\ \end{matrix} . $
For continuity: $ LHS:\underset{x\to 0}{\mathop{\lim }}f(x)=RHS\underset{x\to 0}{\mathop{\lim }}f(x)=f(0) $
$ \Rightarrow \underset{x\to 0}{\mathop{\lim }}f(x)=\underset{x\to 0}{\mathop{\lim }}x^{p}\cos ( \frac{1}{x} )=0 $
$ \Rightarrow \underset{x\to 0}{\mathop{\lim }}x^{p}\cos ( \frac{1}{x} )=0 $
$ \cos ( \frac{1}{x} ) $ is always a finite quantity if $ x\to 0 $
$ \Rightarrow x^{p}=0 $ Which is possible only if $ p>0 $ .
BETA
