1.15 Connecting Limits at Infinity and Horizontal Asymptotes

1.15 Connecting Limits at Infinity and Horizontal Asymptotes

A function can keep changing forever while its output settles toward a fixed value. That “destination” is described by a limit at infinity, and the corresponding graph feature is a horizontal asymptote.

Limits at infinity and end behavior

The statement

$$ \lim_{x\to\infty}f(x)=L $$

means that $f(x)$ approaches $L$ as $x$ increases without bound. Similarly,

$$ \lim_{x\to-\infty}f(x)=L $$

describes what happens as $x$ decreases without bound. These are statements about end behavior, not about the value of the function at any finite input.

Horizontal asymptote: The line $y=L$ is a horizontal asymptote of $f$ if $\lim_{x\to\infty}f(x)=L$ or $\lim_{x\to-\infty}f(x)=L$.

For example, consider

$$ f(x)=\frac{4x^2-7}{2x^2+5}. $$

Divide numerator and denominator by $x^2$:

$$ f(x)=\frac{4-\frac{7}{x^2}}{2+\frac{5}{x^2}}. $$

As $x\to\infty$ or $x\to-\infty$, both $\frac{7}{x^2}$ and $\frac{5}{x^2}$ approach $0$, so

$$ \lim_{x\to\pm\infty}f(x)=\frac{4}{2}=2. $$

Therefore, $y=2$ is a horizontal asymptote.

The same conclusion appears in several representations:

• Analytical: the highest-degree terms give $\frac{4x^2}{2x^2}=2$.
• Numerical: $f(10)=\frac{393}{205}\approx1.917$, while $f(100)\approx1.999$.
• Graphical: the curve approaches the line $y=2$ at both ends.
• Verbal: for very large positive or negative inputs, the output becomes close to $2$.

This is the suggested skill 2.D Connecting Representations: Identify how mathematical characteristics or properties of functions are related in different representations.

The essential knowledge for Topic 1.15 is LIM-2.D.3, which extends limits to infinity; LIM-2.D.4, which connects limits at infinity with end behavior; and LIM-2.D.5, which uses limits to compare relative magnitudes and rates of change. For instance,

$$ \lim_{x\to\infty}\frac{x^2}{e^x}=0 $$

shows analytically that exponential growth eventually dominates quadratic growth. The graph and a sufficiently far-right table should show the ratio approaching $0$ as well.

A related limit appears when adding a finite number of terms from a geometric pattern. If

$$ S_n=1+r+r^2+\cdots+r^{n-1} =\frac{1-r^n}{1-r}, $$

then for $|r|<1$, the term $r^n$ approaches $0$ as $n\to\infty$. Thus,

$$ \lim_{n\to\infty}S_n=\frac{1}{1-r}. $$

The limit describes the eventual value approached by the finite sums; it is not obtained by simply substituting infinity into the formula.

Retrieval check: Find $\lim_{x\to\infty}\frac{5x+1}{x^2+4}$ and name the horizontal asymptote. Dividing every term by $x^2$ gives $\frac{5/x+1/x^2}{1+4/x^2}$, which approaches $\frac{0}{1}=0$. The asymptote is $y=0$: the denominator's higher degree makes it grow faster than the numerator.