The derivative of function f(x ), where f(x) =\\begin{array}{c}{ {\sin \x ^ { 2\end{array} }/\x }, x\neq0; 0, \x=0

Correct Answer is : exists at x = 0 and is also continuous

exists at x = 0 but is not continuous

HCU MCA 2013 Paper

Question : 26 of 75

Marks: +1, -0

The derivative of function f(x ), where
f(x) =

\{\table { {\sin \x ^ { 2 } }/\x }, x≠0; 0, \x=0

Solution:

(b) Given,
f(x) =

\{\table { {\sin \x ^ { 2 } }/\x }, x≠0; 0, \x=0

Now, we check the continuity of f(x) at x = 0, f(0) = 0,
LHL

\;\;\;\;\;\;\;\;\;\;=\f(0-0)=\lim↙{\h→0}\f(0-\h)=\lim↙{\h=0}{\sin \h^2}/{-\h

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=-\lim↙{\h→0}{\sin \h^2}/{\h}^2×\h

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=-\lim↙{\h→0}{\sin \h^2}/{\h}^2×\lim↙{\h→0}\h

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=-(1)×0=0

RHL

\;\;\;\;\;\;\;\;\;\;=\f(0+0)=\lim↙{\h→0}(0+\h)=\lim↙{\h→0}{\sin \h^2}/{\h}

\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\lim↙{\h→0}{\sin \h^2}/{\h}^2×\lim↙{\h→0}\h=1×0=0

∵

\;\;\;\;\;\;'\f(0)=\LHL=\RHL=0

Hence, f(x) exists at x = 0, and is also continuous.

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