Limits at Infinity: Rules, Asymptotes & Examples

What Are Limits at Infinity?

Limits at infinity are the values a function approaches as its input grows without bound, written $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$. They answer one question: once $x$ is enormous, where does $f(x)$ end up? This is called the end behavior of the function, and it is a statement about the far edges of the graph rather than any single point on it.

Two outcomes are possible. The limit can be a finite number $L$, or it can grow without bound and be written as $+\infty$ or $-\infty$.

$$\lim_{x \to \infty} f(x) = L \qquad \text{or} \qquad \lim_{x \to \infty} f(x) = \pm\infty$$

When the limit is a finite number $L$, that value has a geometric name.

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

The two ends can behave differently, so a graph may have one horizontal asymptote on the right and a different one on the left. The single most useful fact for computing these limits is that dividing by a growing power of $x$ drives a term to zero. For the everyday meaning of a limit at a finite point, the limit of a function overview is the companion to this page.

What Is The Key Fact Behind Limits at Infinity?

Everything rests on one reciprocal-power rule. As $x$ grows, any positive power of $x$ in a denominator swamps the numerator, so the fraction collapses to zero.

$$\lim_{x \to \infty} \frac{1}{x^p} = 0 \qquad \text{for any } p > 0$$

The same holds as $x \to -\infty$ whenever $x^p$ is defined there. Geometrically, $y = \tfrac{1}{x^p}$ flattens onto the $x$-axis, so $y = 0$ is its horizontal asymptote. Every rational-function limit in this article is really this one fact applied to each term after a tidy-up step. The standard limits and limit laws pages collect the algebra rules that let you split a limit across sums, products, and quotients before this fact does the work.

How Do You Find Limits at Infinity Of A Rational Function?

For a rational function $\frac{p(x)}{q(x)}$, the end behavior is decided by a race between the highest power on top and the highest power on the bottom. Compare the two degrees, and there are exactly three cases.

Table: End behavior of a rational function as $x \to \pm\infty$, by degree comparison.

In the balanced case, if $p(x)$ has leading term $ax^n$ and $q(x)$ has leading term $bx^n$, the limit is $\frac{a}{b}$. In the top-heavy-by-one case the function has no horizontal asymptote, but polynomial long division reveals a straight slant asymptote it hugs at each end. This connects directly to rational functions and their graphs.

How Do You Divide By The Highest Power Of x?

The reliable technique is to divide every term in the numerator and denominator by the highest power of $x$ that appears in the denominator, then send each reciprocal-power term to zero. This turns a messy fraction into a limit you can read off.

Take the balanced example $\dfrac{3x^2 + 1}{2x^2 - x}$. The highest power on the bottom is $x^2$, so divide every term by $x^2$:

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

Now apply the key fact: $\frac{1}{x^2} \to 0$ and $\frac{1}{x} \to 0$, leaving

$$\lim_{x \to \infty} \frac{3 + 0}{2 - 0} = \frac{3}{2}.$$

The leading coefficients were $3$ and $2$ all along, which is exactly what the balanced case promised. Dividing by the highest power is just a bookkeeping trick that makes the reciprocal-power fact do the deciding.

What Are Some Worked Examples Of Limits at Infinity?

Each example is fully stepped, and every answer is checked against the degree rule that predicts it.

Example 1: A balanced rational function.

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

Divide numerator and denominator by $x^2$:

$$\lim_{x \to \infty} \frac{3 + \frac{1}{x^2}}{2 - \frac{1}{x}} = \frac{3 + 0}{2 - 0} = \frac{3}{2}$$

The degrees are equal, so the rule predicts the ratio of leading coefficients, $\frac{3}{2}$. The two agree.

Final answer: $\displaystyle\lim_{x \to \infty} \frac{3x^2 + 1}{2x^2 - x} = \frac{3}{2}$, so $y = \frac{3}{2}$ is a horizontal asymptote.

Example 2: A bottom-heavy rational function.

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

Divide top and bottom by $x^2$:

$$\lim_{x \to \infty} \frac{\frac{1}{x}}{1 + \frac{1}{x^2}} = \frac{0}{1 + 0} = 0$$

The numerator's degree ($1$) is less than the denominator's ($2$), so the rule predicts $0$.

Final answer: $\displaystyle\lim_{x \to \infty} \frac{x}{x^2 + 1} = 0$, so $y = 0$ is the horizontal asymptote.

Example 3: A top-heavy rational function.

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

Divide top and bottom by $x$, the highest power on the bottom:

$$\lim_{x \to \infty} \frac{x}{1 + \frac{1}{x}} = \frac{\infty}{1 + 0} = \infty$$

The numerator's degree is one higher, so the function grows without bound. Polynomial long division writes $\frac{x^2}{x+1} = x - 1 + \frac{1}{x+1}$, so the graph hugs the slant asymptote $y = x - 1$ at each end while the leftover $\frac{1}{x+1}$ fades to zero.

Final answer: $\displaystyle\lim_{x \to \infty} \frac{x^2}{x + 1} = \infty$; there is no horizontal asymptote, but a slant asymptote $y = x - 1$.

Example 4: A square root, where the sign matters.

Evaluate $\displaystyle\lim_{x \to \infty} \frac{\sqrt{x^2 + 1}}{x}$ and $\displaystyle\lim_{x \to -\infty} \frac{\sqrt{x^2 + 1}}{x}$.

Factor $x^2$ out of the root: $\sqrt{x^2 + 1} = \sqrt{x^2},\sqrt{1 + \frac{1}{x^2}} = |x|\sqrt{1 + \frac{1}{x^2}}$. The absolute value is the whole story. As $x \to \infty$, $x$ is positive so $|x| = x$:

$$\lim_{x \to \infty} \frac{|x|\sqrt{1 + \frac{1}{x^2}}}{x} = \lim_{x \to \infty} \frac{x\sqrt{1 + \frac{1}{x^2}}}{x} = \sqrt{1 + 0} = 1$$

As $x \to -\infty$, $x$ is negative so $|x| = -x$, which flips the sign:

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

Final answer: the limit is $1$ as $x \to \infty$ and $-1$ as $x \to -\infty$. The graph has two horizontal asymptotes, $y = 1$ on the right and $y = -1$ on the left.

Which Functions Grow Fastest At Infinity?

Some functions run to infinity far faster than others, and knowing the pecking order lets you read off many limits without any algebra. For large $x$, with constants $p > 0$ and $a > 1$, the standard ranking is:

$$\ln x ;\ll; x^p ;\ll; a^x ;\ll; x \text{ factorial}$$

where $\ll$ means "grows much slower than," and $x$ factorial means the product $x(x-1)(x-2)\cdots(2)(1)$ for whole-number inputs.

• Logarithms crawl. $\ln x$ heads to infinity, but slower than any positive power of $x$, so $\lim_{x \to \infty} \frac{\ln x}{x} = 0$.

• Powers are the middle gear. Any $x^p$ beats every logarithm but loses to every exponential with base above $1$.

• Exponentials sprint. $a^x$ with $a > 1$ outruns every power, so $\lim_{x \to \infty} \frac{x^{100}}{2^x} = 0$.

• Factorials outrun them all. For whole-number growth, $x$ factorial eventually overtakes any fixed exponential.

Whenever a limit pits two of these against each other in a quotient, the faster-growing one wins: it dominates the fraction, sending it to $0$ or to $\pm\infty$ depending on which side it sits. Some of these standoffs produce an indeterminate form such as $\frac{\infty}{\infty}$, which the hierarchy or a tool like L'Hopital's rule then resolves.

Why Do Limits at Infinity Work?

The idea feels abstract until you picture the graph running off the page. Then it becomes concrete.

• End behavior is a long-run trend. A limit at infinity ignores the wiggles near the origin and reports only where the curve is heading once $x$ is huge. That is why two functions can look completely different up close yet share a horizontal asymptote.

• A horizontal asymptote is a target, not a wall. The line $y = L$ is the height the curve settles toward, but the graph is allowed to touch or cross it. A curve like $\frac{\sin x}{x} + 1$ crosses its asymptote $y = 1$ infinitely often while still converging to it, because the gap keeps shrinking toward zero.

• Dividing by the highest power exposes the dominant term. Every term except the leading ones carries a reciprocal power of $x$ that vanishes. What survives is the race between the fastest-growing pieces, which is precisely what "end behavior" means.

Read this way, the degree rules are not three rules to memorize. They are one idea: the highest power wins, and everything slower fades out of the picture as $x$ grows.

Who Shaped Limits at Infinity?

Infinity was argued over for centuries before anyone pinned it down. The symbol came first, and rigor came much later.

Two named figures anchor the rigor:

• Augustin-Louis Cauchy (1789–1857, France) rebuilt calculus around a careful notion of limit, insisting that convergence be argued, not assumed.

• Karl Weierstrass (1815–1897, Germany) supplied the quantifier definition of a limit that underpins the modern treatment of end behavior and asymptotes.

Where Are Limits at Infinity Used In The Real World?

End behavior answers "what happens in the long run," which is a question almost every field asks.

• Physics: the terminal velocity of a falling object is a limit at infinity of its velocity function, the steady speed it approaches once air resistance balances gravity.

• Biology and medicine: population and dose-response models level off at a carrying capacity or saturation value, the horizontal asymptote of a growth curve.

• Engineering and control: the steady-state response of a circuit or a controller is the limit of its output as time runs on, telling designers where a system finally settles.

• Economics: long-run average cost approaches a floor as output grows, and diminishing-returns models flatten toward a ceiling.

• Computer science: the growth rate of an algorithm's running time is exactly the end behavior of a function, which is why the same degree-and-dominance thinking drives big-O analysis.

One idea, "where is this heading once the inputs are large," unifies terminal velocity, saturation, steady state, and algorithm speed. Reading a graph's far edges is reading its future.

What Are The Most Common Mistakes With Limits at Infinity?

These three errors account for most lost marks on limits at infinity, and each matches a question real students ask on r/learnmath and r/calculus and in course error handouts.

Dropping the sign of $\sqrt{x^2} = |x|$ as $x \to -\infty$.

Where it slips in:

A student simplifies $\sqrt{x^2 + 1}$ to $x\sqrt{1 + \frac{1}{x^2}}$ and gets the limit $1$ at both ends, missing that the left end is $-1$.

Don't do this:

Do not replace $\sqrt{x^2}$ with $x$. The square root of a square is the absolute value, $\sqrt{x^2} = |x|$.

The correct way:

Write $\sqrt{x^2} = |x|$, then split by direction: $|x| = x$ as $x \to \infty$ and $|x| = -x$ as $x \to -\infty$. That sign flip is what turns the left-hand limit into $-1$.

Treating $\frac{\infty}{\infty}$ as something you can plug in.

Where it slips in:

A student substitutes $x = \infty$ into $\frac{3x^2 + 1}{2x^2 - x}$, writes $\frac{\infty}{\infty}$, and either calls it $1$ or calls it undefined and stops.

Don't do this:

Do not read $\frac{\infty}{\infty}$ as a number. It is an indeterminate form, and it can equal any value depending on the two rates.

The correct way:

Resolve it. Divide by the highest power of $x$ (or use the growth hierarchy), which for this fraction gives $\frac{3}{2}$, not $1$.

Believing a graph can never cross its horizontal asymptote.

Where it slips in:

A student rules out an answer because the curve touches or crosses the line $y = L$, assuming an asymptote is a barrier.

Don't do this:

Do not treat a horizontal asymptote as unreachable. Only vertical asymptotes are never crossed; a horizontal one can be crossed, even infinitely often.

The correct way:

Judge by the limit alone. If $\lim_{x \to \infty} f(x) = L$, then $y = L$ is a horizontal asymptote no matter how many times the graph meets it on the way, as $\frac{\sin x}{x} + 1$ does around $y = 1$.

Practice Problems On Limits at Infinity

Work each one, then check against the answer. Answers are verified by the degree rule or the reciprocal-power fact.

1. Evaluate $\displaystyle\lim_{x \to \infty} \frac{5x^3 - 2x}{4x^3 + x^2 + 7}$.
   (Answer: equal degrees, so the ratio of leading coefficients, $\frac{5}{4}$.)

2. Evaluate $\displaystyle\lim_{x \to \infty} \frac{2x + 3}{x^2 - 5}$.
   (Answer: bottom-heavy, so $0$.)

3. Evaluate $\displaystyle\lim_{x \to \infty} \frac{4x^2 - 1}{x + 6}$.
   (Answer: top-heavy by one, so $+\infty$; slant asymptote $y = 4x - 24$.)

4. Evaluate $\displaystyle\lim_{x \to -\infty} \frac{\sqrt{9x^2 + 4}}{x}$.
   (Answer: $\sqrt{9x^2} = 3|x| = -3x$ for $x < 0$, so the limit is $-3$.)

5. Evaluate $\displaystyle\lim_{x \to \infty} \frac{\ln x}{x}$.
   (Answer: $\ln x$ grows slower than $x$, so $0$.)

6. Evaluate $\displaystyle\lim_{x \to \infty} \frac{x^5}{e^x}$.
   (Answer: the exponential outruns the power, so $0$.)

Where Should You Go Next After Limits at Infinity?

Limits at infinity sit at the crossroads of limits and graphing, and several natural doors open from here.

1. Curve sketching. End behavior and asymptotes are two of the ingredients that let you draw a full graph from its equation.

2. Indeterminate forms. Learn how to resolve $\frac{\infty}{\infty}$ and $\frac{0}{0}$ when the growth hierarchy alone does not settle a limit.

3. Limit formulas. Collect the standard results that speed up limits at infinity and elsewhere.

If your child is meeting limits at infinity for the first time, a live Bhanzu trainer teaches it from the end-behavior picture up, so asymptotes and degree rules feel like one idea, in the Bhanzu math program.