NIMCET 2017 — Mathematics PYQ

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Question

NIMCET 2017 — Mathematics PYQ

NIMCET | Mathematics | 2017

The slope of the function $f (x) =< / s p an >< / s t r o n g >< s t r o n g >< s p an s t y l e = " f o n t - s i z e : 14 pt; " > {< / s p an >< / s t r o n g >< s t r o n g >< s p an s t y l e = " f o n t - s i z e : 14 pt; " > x^{2} sin (\frac{1}{x}), < / s p an >< / s t r o n g >< s t r o n g >< s p an s t y l e = " f o n t - s i z e : 14 pt; " > 0, am p; x \neq = 0 am p; x = 0 < / s p an >< / s t r o n g >< s t r o n g >< s p an s t y l e = " f o n t - s i z e : 14 pt; " >$

Choose the correct answer:

Explanation

Concept:
The derivative of a function gives us the slope of the line tangent to the function at any point on the graph.

$\frac{d y}{d x} = lim_{x \to 0} \frac{f ( x ) - f ( a )}{x - a}$

Calculation:

Here, $f (x) =< / s p an >< s p an s t y l e = " f o n t - s i z e : 14 pt; " > {< / s p an >< s p an s t y l e = " f o n t - s i z e : 14 pt; " > x^{2} sin (\frac{1}{x}), < / s p an >< s p an s t y l e = " f o n t - s i z e : 14 pt; " > 0, am p; x \neq = 0 am p; x = 0 < / s p an >< s p an s t y l e = " f o n t - s i z e : 14 pt; " >$

Slope $= m = \frac{d y}{d x} = lim_{x \to 0} \frac{f ( x ) - f ( 0 )}{x - 0}$

$lim_{x \to 0} \frac{x ^{2} s i n ( \frac{1}{x} ) - 0}{x}$

$= lim_{x \to 0} x sin (\frac{1}{x})$

As we know, $- 1 \leq sin (\frac{1}{x}) \leq 1$

Multiplying by $x$ , we get

$\Rightarrow - x \leq x sin (\frac{1}{x}) \leq x$

$\Rightarrow lim_{x \to 0} (- x) \leq lim_{x \to 0} x sin (\frac{1}{x}) \leq lim_{x \to 0} x$

$\Rightarrow 0 \leq lim_{x \to 0} x^{2} sin (\frac{1}{x}) \leq 0$

$∴ lim_{x \to 0} x^{2} sin (\frac{1}{x}) = 0$