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

Solution:
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) 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-2>0} , hence the denominator will be "positive".
Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 2^+}\frac{-5}{x-2}=\frac{\text{negative}}{\text{positive}}=-\infty}

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