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  4. The value of displaystyle limy → ∞ [y sin ((1/y)) - (1/y) ] is equal to

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Q. The value of $\displaystyle \lim_{y \to \infty} \left[y \, sin \left(\frac{1}{y}\right) - \frac{1}{y} \right]$ is equal to

KEAMKEAM 2015Limits and Derivatives

Solution:

$\underset{{y \rightarrow \infty}}{\lim} \left[y \sin \left(\frac{1}{y}\right)-\frac{1}{y}\right]$
$=\underset{{y \rightarrow \infty}}{\lim}\left[\frac{\sin \left(\frac{1}{y}\right)}{\frac{1}{y}}-\frac{1}{y}\right]$
$=1-0=1$