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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
The slope of the function \[
f(x) =
\begin{cases}
x^2 \sin\!\left(\dfrac{1}{x}\right), & \text{if } x \ne 0, \\[8pt]
0, & \text{if } x = 0
\end{cases}
\]
Solution
$f'(0)=\lim_{x\to0}\dfrac{f(x)-f(0)}{x}$
$=\lim_{x\to0}\dfrac{x^2\sin(1/x)}{x}$
$=\lim_{x\to0}x\sin(\frac{1}{x})=0$
For $x\ne0$,
$f'(x)=2x\sin\left(\dfrac{1}{x}\right)-\cos\left(\dfrac{1}{x}\right)$
Answer: $\boxed{f'(0)=0}$ ✅
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