preprekomW

2021-11-06

Find the limits. if the limit does not exist, state it. In each part show your calculations and circle the answer.

$\frac{{x}^{2}-4x+4}{{x}^{2}+x-6}$

Aniqa O'Neill

Skilled2021-11-07Added 100 answers

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.

Left hand limit

The function grows without bound.

Right hand limit

The function decreases without bound.

Since the corresponding two-sided limits are not equal,

Jeffrey Jordon

Expert2022-06-26Added 2605 answers

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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