Slant Asymptote

​Vertical, Horizontal

& Slant Asymptotes​

​Vertical Asymptotes

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​A vertical asymptote is a place where the function is undefined and the limit of the function does not exist. However, you can approach one side of the limit which will either approach ∞ or -∞. The factor that does not cancel from the denominator of the function is where the vertical asymptote occurs on the graph.

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Multiple Choice

Identify all the vertical asymptotes of the function:

$$f\left(x\right)=\frac{x^2+2x-8}{x^2+x-12}$$  

• $$x=-4$$  

• $$x=3$$  

• $$x=4$$  

• $$x=-3$$  

Multiple Choice

Evaluate the limit as $$x\rightarrow3^-$$  of f(x) below:

$$f\left(x\right)=\frac{x^2+2x-8}{x^2+x-12}$$  

•   $$0.5$$  

•   $$\infty$$  

• $$-\infty$$   

•   $$0$$  

Multiple Choice

Evaluate the limit as $$x\rightarrow3^+$$  of f(x) below:

$$f\left(x\right)=\frac{x^2+2x-8}{x^2+x-12}$$  

•   $$0.5$$  

•   $$\infty$$  

• $$-\infty$$   

•   $$0$$  

Multiple Choice

Identify the vertical asymptotes:

$$f\left(x\right)=\frac{x^3+2x^2-24x}{x^2-x}$$   

•    $$x=1$$  

•    $$x=-6$$  

•    $$x=4$$  

•   $$x=0$$   

​Horiztonal Asymptotes

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​To find horizontal asymptotes we simply find the limit as we approach either ∞ or -∞. A function can have at most two horizontal asymptotes in either direction of infinite. If the limit approaches a real number, that is where the horizontal asymptote occurs on the y-axis.

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Multiple Choice

Identify the horizontal asymptote of the function f(x):

$$f\left(x\right)=\frac{x^2-1}{x^2-x-2}$$  

• $$y=-1$$  

• $$y=0$$  

• $$y=2$$  

• $$y=1$$  

Multiple Choice

Identify the horizontal asymptote of the function f(x):

$$f\left(x\right)=\frac{x+2}{\sqrt[]{9x^2+1}}$$  

•   $$y=\frac{1}{3}$$  

•   $$y=3$$  

•   $$y=0.5$$  

•   $$y=0$$  

Multiple Choice

Identify the horizontal asymptote of the function f(x):

$$f\left(x\right)=\frac{x^2+2}{x^3+x^2-1}$$   

•   $$y=0.2$$   

•   $$y=3$$  

•   $$y=0.5$$  

•   $$y=0$$  

​Slant Asymptotes

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​A slant asymptote is neither vertical or horizontal because as x approaches ±∞ the limit is also approaching ±∞. Slant asymptotes are also referred to as another name "oblique" asymptotes.

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Multiple Select

What type of asymptote(s) does this function have?

$$f\left(x\right)=\frac{x^2-6x+7}{x+5}$$  

• Vertical

• Horizontal

• Slant

Multiple Select

What type of asymptote(s) does this function have?

  $$f\left(x\right)=\frac{2x^2-4x+8}{3x^2-27}$$  

• Vertical

• Horizontal

• Slant

Multiple Choice

What type of asymptote(s) does this function have?

   $$f\left(x\right)=\frac{x^3-2x^2+5}{x^2}$$  

• Vertical

• Horizontal

• Slant