Limits at Infinity

Unit 1 · Limits & Continuity

End Behavior and Asymptotes

Limits at infinity describe the end behavior of a function as x grows without bound. For rational functions, compare the degrees of numerator and denominator to find the horizontal asymptote. Vertical asymptotes occur where the denominator is zero and the numerator is not.

Horizontal Asymptotes — Rational Functions

deg(num) < deg(den)

\lim_{x\to\infty} f(x) = 0

deg(num) = deg(den)

\lim_{x\to\infty} f(x) = \frac{\text{leading coeff of num}}{\text{leading coeff of den}}

deg(num) > deg(den)

\lim_{x\to\infty} f(x) = \pm\infty \text{ (no horiz. asymptote)}

Vertical Asymptotes

x = c is a vertical asymptote if

\lim_{x\to c} f(x) = \pm\infty

Caution: If both numerator and denominator vanish at x = c, factor first — it may be a removable hole, not a vertical asymptote.

Type 1

Horizontal Asymptotes

Divide every term by the highest power in the denominator, then let terms with x in the denominator go to zero.

Example 1

Find the limit and state the horizontal asymptote.

\lim_{x \to \infty} \frac{5x^2 - 3x + 1}{2x^2 + 7}

Example 2

Find the limit.

\lim_{x \to -\infty} \frac{6x - 1}{x^2 + 4}

Practice more of this type— AI-generated · infinite problems

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Type 2

Vertical Asymptotes

Find zeros of the denominator. If the numerator is nonzero there, it's a vertical asymptote. If both vanish, factor first.

Example 3

Find all vertical asymptotes.

f(x) = \frac{x+3}{x^2 - x - 6}

Practice more of this type— AI-generated · infinite problems

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