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$\sf \: \lim _{x \longrightarrow - \infty }( \sin( \frac{1}{ \theta} ) )$

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Answer 1

Appropriate Question :-

Evaluate the following limit :-

$\rm :\longmapsto\:\displaystyle\lim_{x \to \: - \: \infty} \: sin \frac{1}{x}$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\displaystyle\lim_{x \to \: - \: \infty} \: sin \frac{1}{x}$

To evaluate this limit, we use method of Substitution

So, Substitute

$\red{\rm :\longmapsto\:x \: = \: - \: \dfrac{1}{y}}$

$\red{\rm :\longmapsto\:As \: x \: \to \: - \infty , \: so \: \: y \: \to \: 0 \: }$

So, above expression

$\rm :\longmapsto\:\displaystyle\lim_{x \to \: - \: \infty} \: sin \frac{1}{x}$

can be rewritten as

$\rm \: = \: \displaystyle\lim_{y \to 0} \: sin( - y)$

$\rm \: = \: - \: \displaystyle\lim_{y \to 0} \: sin( y)$

$\rm \: = \: - \: sin0$

$\rm \: = \: 0$

Hence,

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: \displaystyle\lim_{x \to \: - \: \infty} \: sin \frac{1}{x} = 0 \: }}}$

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