Question

Evaluate lim x —>0 xsin 1/x​

Answer 1

${\LARGE{ \underline{ \text{Basic idea}}}}$

Range of $\sin(x)$ function is Between the closed interval [-1, 1]. That means, doesn't matter what's the input value,

We will use the fact that any constant/finite value when multiplied by 0, will always result 0.

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${\LARGE{\underline{\text{Solution}}}}$

$\implies \lim \limits_{ \:x \to \: 0}x \sin\left( \dfrac{1}{x} \right)$

${ \implies{\lim \limits_{ \:x \to \: 0}x \times \rm\left\{ any \: finite \: value \: between \: - 1 \: and \: 1\right\}}}$

${\implies {0 \times \rm\left\{ any \: finite \: value \: between \: - 1 \: and \: 1\right\}}}$

$\implies \boxed0$

Therefore,

$\boxed{ \lim \limits_{ \:x \to \: 0}x \sin\left( \dfrac{1}{x} \right)= 0}$