Let f : R→ R be defined by f(x) = {cases} x≤ft(1/x), & {if } x>0 \\ 0, & {if } x≤ 0 {cases} . Then

Correct Answer is : f is continuous but not differentiable at 0.

f is neither continuous nor differentiable at 0

f is continuous and differentiable at 0.

f is continuous but not differentiable at 0.

f is not continuous but differentiable at 0.

Correct Answer is : f is continuous but not differentiable at 0.

Test Index

NIMCET 2019 Question Paper with solutions

Question : 2 of 120

Marks: +1, -0

\begin{cases} x\sin\left(\frac{1}{x}\right), & \text{if } x>0 \\ 0, & \text{if } x\le 0 \end{cases}

. Then

Solution:

We have,

f(x) =

\begin{cases} x\sin\left(\frac{1}{x}\right), & \text{if } x>0 \\ 0, & \text{if } x\le 0 \end{cases}

∴(R.H.L at

x=0

)

=\lim\limits_{x\rightarrow 0^{-}}x\sin\left(\frac{1}{x}\right)=\lim\limits_{h\rightarrow 0}h\sin\left(\frac{1}{h}\right)=0\times[-1,1]=0

(LHL at

x=0

) = 0 and

f(0)=0

Hence,f(x) continuous at

x=0

Again,RHD

=\lim\limits_{x\rightarrow 0^{+}}\frac{x\sin\left(\frac{1}{x}\right)-0}{x-0}=\lim\limits_{x\rightarrow 0^{+}}

\sin\left(\frac{1}{x}\right)=\lim\limits_{h\rightarrow 0}\sin\left(\frac{1}{h}\right)=[-1,1]

and LHD = 0

RHD ≠ LHD

Hence,f(x) is not differentiable at

x=0

Go to Question:

Prev Question

Next Question