Limits at infinity and how to find horizontal asymptotes

To find a horizontal asymptote, we look far to the left or far to the right and ask what height the graph approaches. The two ends can tell us different things. Let’s find both limits for one function and use them to read its graph.

What does a limit at infinity tell us?

As x approaches positive infinity, we consider larger and larger positive inputs. As x approaches negative infinity, we consider negative inputs with larger and larger magnitudes. If either end approaches a finite number L, the line y = L is a horizontal asymptote. A graph can cross a horizontal asymptote at a finite input and still approach it at an end.

Example: Find the end behavior of a shifted reciprocal

f (x) = \frac{1}{x - 3}

As $x$ goes far to the right, $x - 3$ grows without bound and its reciprocal approaches $0$ from above. Far to the left, $x - 3$ is negative with arbitrarily large magnitude, so the reciprocal approaches $0$ from below.

\begin{array}{l} \lim_{x \to \infty} \frac{1}{x - 3} = 0 \\ \lim_{x \to - \infty} \frac{1}{x - 3} = 0 \end{array}

Both ends approach y = 0, so the graph has one horizontal asymptote.

The dashed lines mark the horizontal asymptote y = 0 and the vertical asymptote x = 3.

How is a vertical asymptote different?

Near x = 3, the denominator approaches zero. From the right it stays positive, giving an unbounded positive reciprocal. From the left it stays negative, giving an unbounded negative reciprocal.

\begin{array}{l} \lim_{x \to 3^{+}} \frac{1}{x - 3} = \infty \\ \lim_{x \to 3^{-}} \frac{1}{x - 3} = - \infty \end{array}

That gives a vertical asymptote at x = 3. In a limit at infinity the input moves without bound. In these two infinite limits the input approaches a fixed number, and the output moves without bound.

Compare degrees to find a rational function’s horizontal asymptote

Compare the highest powers of the numerator and denominator. The leading terms control the end behavior because the lower powers become small relative to them as $| x |$ grows. Let $a$ and $b$ be the leading coefficients of the numerator and denominator, respectively.

End behavior of a rational function
Degree of numerator	Horizontal asymptote
Less than denominator	$y = 0$
Equal to denominator	$y = \frac{a}{b}$ Divide the leading coefficients.
Greater than denominator	No horizontal asymptote

Example with equal degrees

Both polynomials have degree 2. Divide the numerator and denominator by $x^{2}$ to see what happens as $x$ moves away from zero.

\begin{array}{l} \frac{3 x^{2} + 1}{2 x^{2} - 5} \\ = \frac{3 + \frac{1}{x^{2}}}{2 - \frac{5}{x^{2}}} \end{array}

The terms with $\frac{1}{x^{2}}$ approach zero at both ends. The leading coefficients are $a = 3$ and $b = 2$ , so the horizontal asymptote is

y = \frac{3}{2} .

Learn mathKrista KingApril 10, 2018horizontal asymptotes, limits at infinity