\(f( x ) = \ * 20 c x 2..." - Step-by-step solution, answer key, and detailed explanation on MCA Prep."> The slope of the function at x = 0.
\(f( x ) = \ * 20 c x 2..." - Step-by-step solution, answer key, and detailed explanation on MCA Prep." /> The slope of the function at x = 0.
\(f( x ) = \ * 20 c x 2..." - Step-by-step solution, answer key, and detailed explanation on MCA Prep." />
<p>The slope of the function at x = 0.</p>
<p><span class="math-tex">\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2}\sin \left( {\frac{1}{x}} \right),}&{x \ne 0}\\ {0,}&{x = 0} \end{array}} \right.\)</span> ?</p>
Step-by-step Solution: \[
f(x) =
\begin{cases}
x^2 \sin \left( \frac{1}{x} \right), & x \neq 0 \\
0, & x = 0
\end{cases}
\]
\[
m = \frac{dy}{dx} = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}
\]
\[
\lim_{x \to 0} \frac{x^2 \sin \left( \frac{1}{x} \right) - 0}{x}
\]
\[
\lim_{x \to 0} x \sin \left( \frac{1}{x} \right)
\]
\[
-1 \leq \sin \left( \frac{1}{x} \right) \leq 1
\]
\[
\Rightarrow - x \leq x \sin \left( \frac{1}{x} \right) \leq x
\]
\[
\Rightarrow \lim_{x \to 0} (-x) \leq \lim_{x \to 0} x \sin \left( \frac{1}{x} \right) \leq \lim_{x \to 0} x
\]
\[
\Rightarrow 0 \leq \lim_{x \to 0} x^2 \sin \left( \frac{1}{x} \right) \leq 0
\]
\[
\therefore \lim_{x \to 0} x^2 \sin \left( \frac{1}{x} \right) = 0
\]
Question 38
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