Math 22 Asymptotes

Vertical Asymptotes and Infinite Limits

 If $f(x)$ approaches infinity (or negative infinity) as $x$ approaches Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c}
 from the right or from the left, then the line

Example: Find the limit below:

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

Solution:
Important: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim {\frac {\text{constant}}{0}}} is either $\infty$ or $-\infty$
Notice $x\to 3^{-}$, so $x<3$, then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\to 2^{+}}{\frac {-5}{x-2}}}

Solution:
Notice $x\to 2^{+}$, so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x>2} , then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x-2>0} , hence the denominator will be "positive".
Therefore, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\to 2^{+}}{\frac {-5}{x-2}}={\frac {\text{negative}}{\text{positive}}}=-\infty }

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