Q. 6 - Calculus

Question

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 

Step-by-Step Solution

Answer

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 c and $x=c$ is vertical asymptotes.

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

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.

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

Step 1. Given information.

Vertical Asymptotes 

Horizontal Asymptotes 

Step 2. Description of Vertical 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 c and $x=c$ is vertical asymptotes.

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

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

Step 3. Description of Horizontal Asymptotes

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 $y=0$.