Limits at Infinity and Horizontal Asymptotes

By the end of this lesson, you will be able to evaluate limits as $x \to \pm\infty$ for polynomials, rational functions, radical functions, and exponential/logarithmic functions, and connect these to horizontal and oblique asymptotes.

Limits at Infinity for Polynomials

As $x \to \pm\infty$, only the leading term matters:

$$\lim_{x \to \infty}(a_n x^n + \cdots) = \lim_{x \to \infty}a_n x^n$$

• $n$ even, $a_n > 0$: both $\pm\infty$
• $n$ odd, $a_n > 0$: $-\infty$ and $+\infty$

Limits at Infinity for Rational Functions

$$\lim_{x \to \pm\infty}\frac{a_n x^n + \cdots}{b_m x^m + \cdots}$$

Divide numerator and denominator by $x^m$ (highest power in denominator).

Rules:

• $n < m$: limit = 0 (horizontal asymptote $y = 0$)
• $n = m$: limit = $a_n/b_m$ (horizontal asymptote $y = a_n/b_m$)
• $n > m$: limit = $\pm\infty$ (no horizontal asymptote)

Example 1

$\lim_{x \to \infty}\dfrac{3x^3 - 2x + 5}{2x^3 + x^2 - 1}$

Divide by $x^3$: $= \dfrac{3-2/x^2+5/x^3}{2+1/x-1/x^3} \to \dfrac{3}{2}$

Horizontal asymptote: $y = 3/2$.

Limits Involving Radicals

$\lim_{x \to \infty}\sqrt{4x^2+1}$: factor: $\sqrt{x^2(4+1/x^2)} = x\sqrt{4+1/x^2} \to 2x \to \infty$

$\lim_{x \to \infty}\dfrac{\sqrt{4x^2+1}}{x-3}$:

Divide by $x$ (positive for $x > 0$): $= \dfrac{\sqrt{4+1/x^2}}{1-3/x} \to \dfrac{2}{1} = 2$

For $x \to -\infty$: $\sqrt{x^2} = |x| = -x$ (since $x < 0$), so:

$\dfrac{\sqrt{4+1/x^2}}{1-3/x} \to 2$ but with $|x| = -x$ in the denominator gives $-2$ for $x \to -\infty$.

So: $y = 2$ and $y = -2$ are both horizontal asymptotes.

Exponential and Logarithmic Limits

$$\lim_{x \to \infty}e^x = +\infty, \quad \lim_{x \to -\infty}e^x = 0$$

$$\lim_{x \to \infty}\ln x = +\infty, \quad \lim_{x \to 0^+}\ln x = -\infty$$

Relative rates: $\ln x \ll x^n \ll e^x$ as $x \to \infty$ (exponential grows faster than any polynomial, which grows faster than logarithm).

Example 2 — Mixed

$\lim_{x \to \infty}\dfrac{\ln x}{x}$:

Both grow without bound — indeterminate $\infty/\infty$.