CalcRef

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.

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

Find lim(x→∞) (2x + 1)/(x³ + 4).
  1. Degree of numerator (1) < degree of denominator (3)

Answer: 0

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