what it means, in terms of limits, for a function f to have a vertical asymptote at x = c or a horizontal asymptote at y = L

Vertical Asymptotes

Horizontal Asymptotes

Vertical Asymptotes: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches infinity as x reaches c then we can say that infinity is the limit of $f\left(x\right)$as x approaches cand $x=c$is vertical asymptotes.

$\underset{x\to c}{\lim }f\left(x\right)=\infty$

For example inrole="math" localid="1648276958501" $\underset{x\to 0}{\lim }\frac{1}{x}=\infty$, function value reaches infinity as x reaches zero and vertical asymptotes at$x=0$.

Horizontal Asymptotes: If f is a function and c and L are real numbers where the value of $f\left(x\right)$reaches L as x reaches infinity then we can say that L is the limit of $f\left(x\right)$as x approaches infinity and $y=L$is horizontal asymptotes.

$\underset{x\to \infty }{\lim }f\left(x\right)=L$

For example in$\underset{x\to \infty }{\lim }\frac{1}{x}=0$, function value reaches zero as x reaches infinity and horizontal asymptotes at$y=0$.