Asymptote Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Asymptote.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An asymptote is a line that a curve approaches arbitrarily closely as the input (or output) grows without bound, but typically never reaches.

The graph gets infinitely close but never touches—like chasing something forever.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Asymptotes describe behavior at extremes (x \to \infty or x \to some value).

Common stuck point: Graphs can cross horizontal asymptotes—they just approach them at infinity.

Sense of Study hint: Ask yourself: what happens to y as x gets extremely large or extremely small? Plug in x = 1000 and x = -1000 to see the trend.

Worked Examples

Example 1

easy
Find the vertical and horizontal asymptotes of f(x) = \dfrac{3x}{x - 2}.

Solution

  1. 1
    Vertical asymptote: set the denominator equal to zero. x - 2 = 0 \Rightarrow x = 2. Since the numerator 3(2)=6 \neq 0, there is a vertical asymptote at x = 2.
  2. 2
    Horizontal asymptote: compare degrees. Both numerator and denominator have degree 1. Divide leading coefficients: \frac{3}{1} = 3. Thus the horizontal asymptote is y = 3.
  3. 3
    Verify by taking the limit: \lim_{x\to\infty}\frac{3x}{x-2} = \lim_{x\to\infty}\frac{3}{1-2/x} = 3.

Answer

Vertical asymptote: x = 2; Horizontal asymptote: y = 3
Vertical asymptotes arise where the denominator is zero (and numerator is not). Horizontal asymptotes describe end behavior; when degrees are equal, the asymptote equals the ratio of leading coefficients.

Example 2

hard
Find the oblique (slant) asymptote of g(x) = \dfrac{x^2 + x - 1}{x - 1}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Identify all asymptotes of h(x) = \dfrac{5}{x + 3}.

Example 2

medium
For f(x) = \dfrac{2x^2 - 3}{x^2 + 1}, determine the horizontal asymptote and verify by computing f(100).
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Related Concepts

FunctionLimitInfinity

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition