Math 22 Asymptotes

Vertical Asymptotes and Infinite Limits

 If $f(x)$ approaches infinity (or negative infinity) as $x$ approaches $c$ from the right or from the left, then the line

Example: Find the limit below:

1) $\lim _{x\to 3^{-}}{\frac {-2}{x-3}}$

ExpandSolution:
Important: $\lim {\frac {\text{constant}}{0}}$ is either $\infty$ or $-\infty$
Notice $x\to 3^{-}$, so $x<3$, then $x-3<0$, hence the denominator will be "negative".
Therefore, $\lim _{x\to 3^{-}}{\frac {-2}{x-3}}={\frac {\text{negative}}{\text{negative}}}=\infty$

2) $\lim _{x\to 2^{+}}{\frac {-5}{x-2}}$

ExpandSolution:
Notice $x\to 2^{+}$, so $x>2$, then $x-2>0$, hence the denominator will be "positive".
Therefore, $\lim _{x\to 2^{+}}{\frac {-5}{x-2}}={\frac {\text{negative}}{\text{positive}}}=-\infty$

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