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Limits & Continuity · Continuity & Theorems

Limits at Infinity

lim⁡x→∞anxn+⋯bmxm+⋯={0n<man/bmn=m±∞n>m\lim_{x \to \infty} \frac{a_n x^n + \cdots}{b_m x^m + \cdots} = \begin{cases} 0 & n < m \\ a_n/b_m & n = m \\ \pm\infty & n > m \end{cases}

For rational functions as x→∞, compare the degrees of numerator (n) and denominator (m) to determine the limit.

  • 2 worked examples
  • 18 in Limits & Continuity

Worked examples

Find lim(x→∞) (3x² + 1)/(5x² - 2).
  1. Degrees are equal (both 2)
  2. Limit = leading coefficients ratio = 3/5

Answer: 3/5

1 more worked examplePremium

Find lim(x→∞) (2x + 1)/(x³ + 4).

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Common questions

What is the Limits at Infinity?

For rational functions as x→∞, compare the degrees of numerator (n) and denominator (m) to determine the limit. It is one of the limits & continuity formulas in the CalcRef reference.

How do you use the Limits at Infinity?

Worked example. Find lim(x→∞) (3x² + 1)/(5x² - 2). Degrees are equal (both 2). Limit = leading coefficients ratio = 3/5. Answer: 3/5.

Practice

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