# Find the limits: lim_{xrightarrow3}frac{-2}{x-3}

Find the limits:
$\underset{x\to 3}{lim}\frac{-2}{x-3}$
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Obiajulu
Given:
$\underset{x\to 3}{lim}\left(-\frac{2}{x-3}\right)$
LEFT-HAND LIMIT
Apply the constant multiple rule
$\underset{x\to {3}^{-}}{lim}\left(-\frac{2}{x-3}\right)=\left(-2\underset{x\to {3}^{-}}{lim}\frac{1}{x-3}\right)$
The function decreases without a bound:
$\underset{x\to {3}^{-}}{lim}\frac{1}{x-3}=-\mathrm{\infty }$
Therefore,
$\underset{x\to {3}^{-}}{lim}\left(-\frac{2}{x-3}\right)=\mathrm{\infty }$
RIGHT-HAND LIMIT
Apply the constant multiple rule
$\underset{x\to {3}^{+}}{lim}\left(-\frac{2}{x-3}\right)=\left(-2\underset{x\to {3}^{+}}{lim}\frac{1}{x-3}\right)$
The function grows without a bound:
$\underset{x\to {3}^{+}}{lim}\frac{1}{x-3}=\mathrm{\infty }$
Therefore,
$\underset{x\to {3}^{+}}{lim}\left(-\frac{2}{x-3}\right)=-\mathrm{\infty }$
Answer: $\underset{x\to 3}{lim}\left(-\frac{2}{x-3}\right)$ does not exist, since the corresponding one-sided limits are not equal:

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