Limit at ±∞: keep only the dominant powers

The golden rule

At $\pm\infty$, a polynomial has the same limit as its highest-degree term. A rational function has the same limit as the quotient of its highest-degree terms.

Polynomials

$\displaystyle \lim_{x \to +\infty} (2x^3 - 5x^2 + 100x + 7) = \lim_{x \to +\infty} 2x^3 = +\infty$

The $-5x^2$, $+100x$ and $+7$ are nothing next to $2x^3$.

Rational functions

$\displaystyle \lim_{x \to +\infty} \dfrac{4x^2 + 3x - 1}{2x^2 + 7} = \lim_{x \to +\infty} \dfrac{4x^2}{2x^2} = 2$

Three cases based on numerator degree ($n$) and denominator degree ($d$):

Case	Limit at $+\infty$
$n < d$	$0$
$n = d$	ratio of leading coefficients
$n > d$	$\pm\infty$ (based on signs)

Careful at $-\infty$ for odd degrees

$\displaystyle \lim_{x \to -\infty} x^3 = -\infty$ (not $+\infty$). Odd powers keep the sign of $x$.

Where this trick doesn’t work

• At a finite value (there you factor differently).
• When exponentials or logarithms are involved — their growth beats polynomials.

?Your turn

lim (x → +∞) (3x⁴ − x² + 10) / (x⁴ + 5) is:

A.0B.3C.+∞D.1/3

Check