 # Find the limits. if the limit does not exist, state it. In each part show your c preprekomW 2021-11-06 Answered
Find the limits. if the limit does not exist, state it. In each part show your calculations and circle the answer.
$\frac{{x}^{2}-4x+4}{{x}^{2}+x-6}$
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Given:
$f\left(x\right)=\frac{{x}^{2}-4x+4}{{x}^{2}+x-6}$
Left hand limit
$\underset{x\to -{3}^{-}}{lim}f\left(x\right)=\underset{x\to -{3}^{-}}{lim}\frac{{x}^{2}-4x+4}{{x}^{2}+x-6}$
$\underset{x\to -{3}^{-}}{lim}f\left(x\right)=\mathrm{\infty }$
The function grows without bound.
Right hand limit
$\underset{x\to -{3}^{+}}{lim}f\left(x\right)=\underset{x\to -{3}^{+}}{lim}\frac{{x}^{2}-4x+4}{{x}^{2}+x-6}$
$\underset{x\to -{3}^{+}}{lim}f\left(x\right)=-\mathrm{\infty }$
The function decreases without bound.
$\underset{x\to -{3}^{-}}{lim}f\left(x\right)\ne \underset{x\to -{3}^{+}}{lim}f\left(x\right)$
Since the corresponding two-sided limits are not equal, $\underset{x\to -3}{lim}\frac{{x}^{2}-4x+4}{{x}^{2}+x-6}$ does not exist.
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