Evaluate the limit: lim_{x rightarrow infty} | vedclass

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(C) To evaluate the limit $\lim_{x \rightarrow \infty} x \sin \left(\frac{2}{x}\right)$,let $t = \frac{1}{x}$.As $x \rightarrow \infty$,$t \rightarrow

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$2$

Vedclass · Answer

(C) To evaluate the limit $\lim_{x ightarrow \infty} x \sin \left(\frac{2}{x}ight)$,let $t = \frac{1}{x}$.As $x ightarrow \infty$,$t ightarrow 0$.Substituting $x = \frac{1}{t}$ into the expression,we get:$\lim_{t ightarrow 0} \frac{1}{t} \sin(2t)$$= \lim_{t ightarrow 0} \frac{\sin(2t)}{t}$$= \lim_{t ightarrow 0} 2 \cdot \frac{\sin(2t)}{2t}$Since $\lim_{	heta ightarrow 0} \frac{\sin 	heta}{	heta} = 1$,we have:$= 2 \cdot 1 = 2$

Evaluate the limit: $\lim_{x \rightarrow \infty} x \sin \left(\frac{2}{x}\right)$

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