Horizontal Asymptotes | Tony's Intertwingled Memex

Core Definition

The horizontal line $y = L$ is a horizontal asymptote of the graph of $y = f(x)$ if either:

\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L

Concept Explanation

A horizontal asymptote represents the baseline boundary value that the outputs of a function approach as the input heads towards positive or negative infinity.

A single function can have:

No horizontal asymptotes (e.g., $f(x) = x^3$ ).
One horizontal asymptote shared on both sides (e.g., $f(x) = \frac{2x}{x+1} \implies y = 2$ ).
Two distinct horizontal asymptotes (e.g., $f(x) = \arctan(x) \implies y = \frac{\pi}{2}, y = -\frac{\pi}{2}$ , or rational radical functions like $f(x) = \frac{3x}{\sqrt{x^2+1}} \implies y = 3, y = -3$ ).

        y ^
      L_1 + - - - - - - - - - - - - - - - - (Right-side HA y = L_1)
          |              . ~ ~ ~ ~ ~ - -
          |         . '
      L_2 + - - - - - - - - - - - - - - - - (Left-side HA y = L_2)
          +--------------------------------> x

Unlike vertical asymptotes (which a continuous function can never cross near the asymptote value), a function can cross its horizontal asymptote multiple times (or infinitely many times, like $f(x) = \frac{\sin(x)}{x}$ ) for finite values of $x$ . A horizontal asymptote only constrains the limiting behavior at the extremes of the domain.

Horizontal Asymptote Rules for Rational Functions

For a rational function $f(x) = \frac{P(x)}{Q(x)} = \frac{a_n x^n + \dots + a_0}{b_m x^m + \dots + b_0}$ :

Degree Comparison	Growth Relationship	Limit as $x \to \pm\infty$	Horizontal Asymptote
$n < m$ (Bottom-Heavy)	Denominator dominates	$\lim_{x \to \pm\infty} f(x) = 0$	$y = 0$ ( $x$ -axis)
$n = m$ (Balanced)	Equal growth rates	$\lim_{x \to \pm\infty} f(x) = \frac{a_n}{b_m}$	$y = \frac{a_n}{b_m}$
$n > m$ (Top-Heavy)	Numerator dominates	$\lim_{x \to \pm\infty} f(x) = \pm\infty$	None (Slant / Oblique if $n = m+1$ )

Example Problems

Example Problem 1: Rational Function with Equal Degrees

Find all horizontal asymptotes for $f(x) = \frac{6x^2 - 5x + 1}{2x^2 + 7}$ .

Solution

Step 1: Compare the highest degrees
Degree of numerator $= 2$ , degree of denominator $= 2$ . Balanced degrees ( $n = m = 2$ ).

Step 2: Evaluate the limit at infinity

\lim_{x \to \pm\infty} \frac{6x^2 - 5x + 1}{2x^2 + 7} = \frac{6}{2} = 3

Conclusion: The function has a single horizontal asymptote at $y = 3$ .

Example Problem 2: Function with Two Distinct Horizontal Asymptotes

Find all horizontal asymptotes for $g(x) = \frac{5x + 2}{\sqrt{4x^2 + 1}}$ .

Solution

Step 1: Evaluate limit as $x \to \infty$ ( $x > 0 \implies \sqrt{x^2} = x$ )
Divide numerator and denominator by $x$ :

\lim_{x \to \infty} \frac{5 + \frac{2}{x}}{\sqrt{4 + \frac{1}{x^2}}} = \frac{5 + 0}{\sqrt{4 + 0}} = \frac{5}{2}

Right-hand horizontal asymptote: $y = \frac{5}{2}$ .

Step 2: Evaluate limit as $x \to -\infty$ ( $x < 0 \implies \sqrt{x^2} = |x| = -x$ )
For $x < 0$ , dividing inside the square root by $x^2$ is equivalent to dividing outside by $-x$ :

\lim_{x \to -\infty} \frac{5x + 2}{\sqrt{4x^2 + 1}} = \lim_{x \to -\infty} \frac{5 + \frac{2}{x}}{-\sqrt{4 + \frac{1}{x^2}}} = \frac{5 + 0}{-\sqrt{4 + 0}} = -\frac{5}{2}

Left-hand horizontal asymptote: $y = -\frac{5}{2}$ .

Conclusion: $g(x)$ has two distinct horizontal asymptotes: $y = \frac{5}{2}$ and $y = -\frac{5}{2}$ .

Connections & References

Parent Concepts: Limits at Infinity, Limits
Related Concepts: Vertical Asymptotes, Growth Dominance in Limits, Limits with Radicals and Conjugates
Course Links: Calc-1 Session 04, Calc-1 Session 16