4.6 Limits at Infinity and Asymptotes

Big idea

Earlier limits asked what happens as the input approaches a number. This section asks what happens as the input runs off without bound. The answer describes the far ends of a graph, which is exactly the part a sketch needs and the part a formula hides.

There are only three possibilities at each end. The values can settle toward a single number, which draws a horizontal asymptote. They can grow without bound in one direction or the other. Or they can do neither, oscillating forever like $\sin x$. Deciding which happens is usually a matter of finding the dominant term and dividing it out.

Vertical asymptotes come from the opposite situation: a finite input where the output blows up, typically a zero of a denominator that is not cancelled by a matching zero on top. The two kinds of asymptote answer different questions and are found by different computations, so keep them separate in your head even though a single graph often has both.

Assembling everything from this chapter gives a curve sketching procedure. Domain and intercepts frame the picture, asymptotes pin down the boundaries, the first derivative marks the rises and falls, and the second derivative fixes the bending. A sketch built this way is not an approximation to a plotted picture; it records the features the plot was made of.

Decoder

The line $y = L$ is a horizontal asymptote of $f$ if the values of $f(x)$ approach $L$ as $x$ grows without bound in the positive or the negative direction.

The two directions are separate claims. A function may have one horizontal asymptote, two different ones, or none, and checking one end tells you nothing about the other. Functions built from even roots are the usual source of two distinct answers, because $\sqrt{x^2} = |x|$ rather than $x$, and the absolute value flips sign when $x$ is negative.

“Approach” is not “never reach.” A graph may cross its horizontal asymptote any number of times; the asymptote only constrains the far end. A damped oscillation crosses its horizontal asymptote infinitely often and still has one.

The classic mistake is dropping the absolute value when you pull a variable out of a square root. Writing $\sqrt{4x^2 + 3} = x\sqrt{4 + 3/x^2}$ is right only for $x > 0$. For $x < 0$ the correct factoring carries a minus sign, and forgetting it hands you the wrong asymptote at one end.

Definitions and results

Limit at infinity. The statement $\lim_{x \to \infty} f(x) = L$ means the values of $f(x)$ can be kept as close to $L$ as desired by taking $x$ large enough. The definition for $x \to -\infty$ is the mirror image.

Horizontal asymptote. The line $y = L$ is a horizontal asymptote of the graph of $f$ if $\lim_{x\to\infty} f(x) = L$ or $\lim_{x\to-\infty} f(x) = L$. A graph has at most two.

Vertical asymptote. The line $x = a$ is a vertical asymptote if at least one of the one sided limits of $f$ at $a$ is $\infty$ or $-\infty$. For a rational function in lowest terms, these occur exactly at the zeros of the denominator.

The basic reciprocal limits. For any $r > 0$,

$$ \lim_{x\to\infty}\frac{1}{x^{r}} = 0, \qquad \lim_{x\to-\infty}\frac{1}{x^{r}} = 0 $$

whenever $x^{r}$ is defined. Every rational limit at infinity reduces to these.

Method for rational functions. Divide numerator and denominator by the highest power of $x$ that appears in the denominator, then take the limit term by term. The three outcomes: if the degrees are equal, the limit is the ratio of the leading coefficients; if the numerator has smaller degree, the limit is $0$; if the numerator has larger degree, the values grow without bound.

Square roots and the sign of $x$. Since $\sqrt{x^2} = |x|$, factoring out of a root gives $\sqrt{ax^2 + b} = |x|\sqrt{a + b/x^{2}}$, which equals $x\sqrt{a + b/x^2}$ for $x > 0$ and $-x\sqrt{a + b/x^2}$ for $x < 0$.

Slant asymptote. If the degree of the numerator is exactly one more than that of the denominator, polynomial division writes the function as a linear polynomial plus a remainder term that tends to $0$. The graph approaches the line $y = mx + b$ at both ends, and that line is a slant asymptote.

Curve sketching checklist. Domain and intercepts; symmetry; vertical asymptotes at the bad points of the domain; end behavior and horizontal or slant asymptotes; intervals of increase and decrease with local extrema from $f'$; concavity and inflection points from $f''$. Plot the special points last and connect them so that all of the above stay true.

Worked examples

Equal degrees

Evaluate $\displaystyle\lim_{x\to\infty} \frac{3x^2 - 2x + 1}{5x^2 + 4}$.

The highest power in the denominator is $x^2$, so divide every term by $x^2$:

$$ \frac{3x^2 - 2x + 1}{5x^2 + 4} = \frac{3 - \dfrac{2}{x} + \dfrac{1}{x^{2}}}{5 + \dfrac{4}{x^{2}}} $$

Each reciprocal power tends to $0$, leaving $3/5$. The same computation works as $x \to -\infty$, since the reciprocal powers vanish at both ends, so $y = 3/5$ is the only horizontal asymptote. The shortcut is the ratio of leading coefficients, and the division above is why that shortcut is legitimate.

A root that gives two different asymptotes

Find all horizontal asymptotes of $\displaystyle f(x) = \frac{\sqrt{4x^2+3}}{2x - 1}$.

For $x > 0$, divide top and bottom by $x$, writing the top division as a division of the radicand by $x^2$:

$$ \frac{\sqrt{4x^2+3}}{2x-1} = \frac{\sqrt{4 + 3/x^{2}}}{2 - 1/x} \longrightarrow \frac{\sqrt{4}}{2} = 1 $$

For $x < 0$ the same division brings a sign out of the root, because $\sqrt{x^2} = -x$ there:

$$ \frac{\sqrt{4x^2+3}}{2x-1} = \frac{-\sqrt{4 + 3/x^{2}}}{2 - 1/x} \longrightarrow \frac{-2}{2} = -1 $$

So there are two horizontal asymptotes, $y = 1$ on the right and $y = -1$ on the left. Sanity check with a large negative input: at $x = -100$ the numerator is about $200$, positive as every square root is, and the denominator is about $-201$, so the quotient is near $-1$.

A slant asymptote

Describe the asymptotes of $\displaystyle f(x) = \frac{x^2+1}{x-1}$.

The denominator vanishes at $x = 1$ while the numerator equals $2$ there, so $x = 1$ is a vertical asymptote. Approaching from the right the denominator is small and positive, so the values run to $\infty$; from the left they run to $-\infty$.

For the end behavior, divide:

$$ \frac{x^2+1}{x-1} = x + 1 + \frac{2}{x-1} $$

Check the division by multiplying back: $(x+1)(x-1) + 2 = x^2 - 1 + 2 = x^2 + 1$. The remainder term tends to $0$ at both ends, so the graph approaches the line $y = x + 1$. There is no horizontal asymptote, since the values grow without bound.

A complete sketch

Sketch $\displaystyle f(x) = \frac{x^2}{x^2 - 1}$.

Domain: all $x$ except $\pm 1$. The only intercept is the origin. Replacing $x$ by $-x$ leaves the formula unchanged, so the graph is symmetric about the $y$ axis.

Asymptotes: the denominator vanishes at $x = \pm 1$ and the numerator does not, so both are vertical asymptotes. Dividing by $x^2$ gives $1/(1 - 1/x^2) \to 1$ at both ends, so $y = 1$ is a horizontal asymptote.

Derivative:

$$ f'(x) = \frac{2x(x^2-1) - x^2(2x)}{(x^2-1)^2} = \frac{-2x}{(x^2-1)^2} $$

The denominator is positive wherever it is defined, so $f'$ has the sign of $-2x$: the function increases for $x < 0$ and decreases for $x > 0$, apart from the breaks at $\pm 1$. The only critical point is $x = 0$, a local maximum with value $0$.

Assemble: on $(-1,1)$ the graph rises from $-\infty$ to the peak at the origin and falls back to $-\infty$. On $(1,\infty)$ it drops from $+\infty$ toward the line $y = 1$ from above, and the mirror image happens on $(-\infty,-1)$. Test one value to confirm: $f(2) = 4/3$, above $1$ as predicted, and $f(0.5) = 0.25/(-0.75) = -1/3$, below the axis as predicted.

Practice

Start with the limits themselves. These ask for end behavior of rational and algebraic expressions, including the cases where the two ends disagree.

Then the whole picture. These ask you to combine asymptotes with the first and second derivative information into a description or a sketch of the graph.

Quiz

Five items on limits at infinity, horizontal and vertical asymptotes, slant asymptotes, and reading a graph from derivative and asymptote data.