# Asymptotes - Video Tutorials & Practice Problems

## Introduction to Asymptotes

Sketch the graph of the function $f\left(x\right)=\frac{1}{x^2}$. Identify the asymptotes on the graph.

Vertical Asymptote: $x=0$, Horizontal Asymptote: None

Vertical Asymptote: $x=0$, Horizontal Asymptote: $y=0$

Vertical Asymptote: $x=0$, Horizontal Asymptote: $y=0$

Vertical Asymptote: $x=1$ , Horizontal Asymptote: $y=0$

## Determining Vertical Asymptotes

Based only on the vertical asymptotes, which of the following graphs could be the graph of the given function? $f\left(x\right)=\frac{x^2-4x}{x^2-x-12}$

## Determining Removable Discontinuities (Holes)

Find all vertical asymptotes and holes of each function. $f\left(x\right)=\frac{-5x}{\left(2x-3\right)^2}$

Hole(s): $x=0$ , Vertical Asymptote(s): $x=\frac32$

Hole(s): $x=\frac32$ , Vertical Asymptote(s): $x=\frac32$

Hole(s): $x=0$ , Vertical Asymptote(s): $x=0$

Hole(s): None , Vertical Asymptote(s): $x=\frac32$

Find all vertical asymptotes and holes of each function. $f\left(x\right)=\frac{x^2-2x}{2x^3-x^2-6x}$

Hole(s): None, Vertical Asymptote(s): $x=0,x=2,x=-\frac32$

Hole(s): $x=0$, Vertical Asymptote(s): $x=-\frac32$

Hole(s): $x=0$, $x=2$, Vertical Asymptote(s): $x=-\frac32$

Hole(s): $x=0$, $x=2$, Vertical Asymptote(s): $x=\frac32$

Find all vertical asymptotes and holes of each function. $f\left(x\right)=\frac{x^2+10x+25}{2x^2+8x-10}$

Hole(s): None, Vertical Asymptote(s): $x=-5,$ $x=1$

Hole(s): $x=-5$ , Vertical Asymptote(s): $x=1$

Hole(s): $x=1$ , Vertical Asymptote(s): $x=-5$

Hole(s): $x=-5$ , Vertical Asymptote(s): $x=-1$

## Determining Horizontal Asymptotes

Find the horizontal asymptote of each function. $f\left(x\right)=\frac{-5x}{\left(2x+3\right)^2}$

Horizontal Asymptote at $y=0$

Horizontal Asymptote at $y=-\frac54$

Horizontal Asymptote at $y=-\frac52$

Find the horizontal asymptote of each function. $f\left(x\right)=\frac{8x^2+1}{2x^2-x-6}$

Horizontal Asymptote at $y=0$

Horizontal Asymptote at $y=\frac14$

Horizontal Asymptote at $y=4$

Find the horizontal asymptote of each function. $f\left(x\right)=\frac{x^2+4x}{2x^3+8x^2}$

Horizontal Asymptote at $y=0$

Horizontal Asymptote at $y=\frac12$

Horizontal Asymptote at $y=2$

## Do you want more practice?

- In Exercises 1–8, find the domain of each rational function. f(x)=5x/(x−4)
- In Exercises 1–8, find the domain of each rational function. g(x)=3x^2/(x−5)(x+4)
- In Exercises 1–8, find the domain of each rational function. h(x)=(x+7)/(x^2−49)
- Use the graphs of the rational functions in choices A–D to answer each question. There may be more than one co...
- Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. As x...
- Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. As x...
- Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20. As ...
- Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II c...
- In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, ...
- Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II c...
- In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, ...
- In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, ...
- Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function....
- Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function....
- Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function....
- In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. h(x)=1/x + ...
- In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+1...
- Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach...
- In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+2...
- Work each problem. Which function has a graph that does not have a horizontal asymptote? A. ƒ(x)=(2x-7)/(x+3)...
- Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.
- Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.
- Graph each rational function. See Examples 5–9. ƒ(x)=(x+1)/(x-4)
- Graph each rational function. See Examples 5–9. ƒ(x)=3x/(x^2-x-2)
- In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x^2+x−6)
- In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x^2−4)
- Solve each problem. Work each of the following. Find an equation for a possible corresponding rational functio...
- Solve each problem. Work each of the following. Find an equation for a possible corresponding rational functio...
- In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=x^4/(x^2+2)
- Solve each problem. Find a rational function ƒ having the graph shown.
- In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the...
- In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the...
- In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the...