Blog · Calculus guides · 4 November 2026 · 4 min read

By the Asymptote team. Every answer on this page is checked by the same machine that grades the game.

Limits at Infinity: Horizontal Asymptotes & Practice Problems

A limit at infinity asks what $f (x)$ settles towards as $x$ grows without bound. For a rational function, compare the highest powers: if the bottom's is bigger, the limit is $0$ ; if they are equal, it is the ratio of the leading coefficients; if the top's is bigger, there is no finite limit. A finite limit $L$ means the graph has a horizontal asymptote $y = L$ .

The formula

x \to \infty lim \frac{a _{n} x ^{n} + \dots}{b _{m} x ^{m} + \dots} = ⎩ ⎨ ⎧ 0 \frac{a _{n}}{b _{m}} \pm \infty n < m n = m n > m

At a large enough $x$ , only the leading terms matter. Everything else is a rounding error with ideas above its station.

When to use it

Use these rules when $x \to \infty$ or $x \to - \infty$ , to find the end behavior of a function or its horizontal asymptotes. For rational functions the degree rules are instant. With square roots, divide by the highest power carefully and watch the sign when $x$ is negative. And for a difference of two large things, like $x^{2} + 6 x - x$ , multiply by the conjugate first. If you end up with $\frac{\infty}{\infty}$ that none of these tidy, L'Hôpital's rule also works at infinity.

How to do it, step by step

Find the highest power of $x$ in the denominator. For $x^{2} + \dots$ , that counts as $x^{1}$ .
Divide every term, top and bottom, by that power. This changes nothing about the value, only how it looks.
Send each $\frac{c}{x ^{k}}$ to $0$ . Any constant over a positive power of $x$ vanishes as $x \to \pm \infty$ .
Read off what is left. A number is the limit and the horizontal asymptote; anything still growing means the limit is $\infty$ or $- \infty$ .
Check $- \infty$ separately when there are roots. For $x < 0$ , $x^{2} = ∣ x ∣ = - x$ , which can flip the sign of the answer.

Worked examples

Example 1: Equal degrees

Easy

Find the limit:

x \to \infty lim \frac{3 x ^{2} - 5 x + 1}{2 x ^{2} + 7}

Top and bottom are both degree 2. Divide every term by $x^{2}$ :

\frac{3 - \frac{5}{x} + \frac{1}{x ^{2}}}{2 + \frac{7}{x ^{2}}} ⟶ \frac{3 - 0 + 0}{2 + 0}

For large $x$ , the $- 5 x$ , the $+ 1$ and the $+ 7$ are tiny next to the squared terms, and dividing by $x^{2}$ is just a way of making that precise. What survives is the ratio of the leading coefficients, as the rule promised. The graph has a horizontal asymptote at $y = \frac{3}{2}$ , at both ends.

Answer

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

Example 2: A square root on top

Medium

Find the limit, then both horizontal asymptotes:

x \to \infty lim \frac{4 x ^{2} + 1}{3 x - 5}

The top behaves like $4 x^{2} = 2∣ x ∣$ , so it is degree 1, like the bottom. Divide by $x$ . For $x > 0$ , $x = x^{2}$ , so the $x$ can go inside the root as $x^{2}$ :

\frac{4 + \frac{1}{x ^{2}}}{3 - \frac{5}{x}} ⟶ \frac{4}{3} = \frac{2}{3} (x \to \infty)

As $x \to - \infty$ , the bottom is negative but the root is still positive, so the same working gives $- \frac{2}{3}$ . Two different horizontal asymptotes, one at each end: $y = \frac{2}{3}$ and $y = - \frac{2}{3}$ . A rational function can never do this, which is why the square root deserves suspicion.

Answer

x \to \infty lim \frac{4 x ^{2} + 1}{3 x - 5} = \frac{2}{3}

Example 3: Infinity minus infinity

Harder

Find the limit:

x \to \infty lim (x^{2} + 6 x - x)

Both terms grow without bound, and $\infty - \infty$ could be anything. Multiply by the conjugate over itself to turn the difference into a fraction:

(x^{2} + 6 x - x) \cdot \frac{x ^{2} + 6 x + x}{x ^{2} + 6 x + x} = \frac{( x ^{2} + 6 x ) - x ^{2}}{x ^{2} + 6 x + x} = \frac{6 x}{x ^{2} + 6 x + x}

Now divide top and bottom by $x$ (positive, so it goes inside the root as $x^{2}$ ):

\frac{6}{1 + \frac{6}{x} + 1} ⟶ \frac{6}{1 + 1}

So $x^{2} + 6 x$ ends up exactly $3$ more than $x$ , no matter how big $x$ gets. The guess "both are about $x$ , so the answer is $0$ " misses it completely.

Answer

x \to \infty lim (x^{2} + 6 x - x) = 3

Limits at infinity practice problems

Three to try, easy to harder. Type an answer and it is checked instantly by the same checker that grades the game; any correct form counts. Stuck? The answer and the first step are under each one.

Problem 1

Easy

Find the limit:

x \to \infty lim \frac{4 x ^{2} - 3 x + 1}{6 x ^{2} + 5}

Stuck? Show the answer

Answer

\frac{2}{3}

How to start

Equal degrees, so only the leading coefficients matter: divide every term by $x^{2}$ and let the rest vanish.

Problem 2

Medium

Find the limit (note the minus sign):

x \to - \infty lim \frac{4 x ^{2} + 3}{5 x - 2}

Stuck? Show the answer

Answer

- \frac{2}{5}

How to start

For $x < 0$ , $x^{2} = - x$ , so dividing the root by $x$ puts a minus sign in front: $\frac{4 x ^{2} + 3}{x} = - 4 + \frac{3}{x ^{2}}$ .

Problem 3

Harder

Find the limit:

x \to \infty lim (x^{2} + 5 x - x^{2} - 3 x)

Stuck? Show the answer

Answer

4

How to start

Multiply by the conjugate over itself. The top becomes $(x^{2} + 5 x) - (x^{2} - 3 x) = 8 x$ ; then divide top and bottom by $x$ .

Want more? Asymptote has 500 more limits at infinity problems, free, with no account.

Common mistakes

Comparing the wrong coefficients. The rule uses the leading coefficients, the ones on the highest power, not the first numbers written. In $\frac{1 + 2 x ^{2}}{5 x ^{2} - x}$ the answer is $\frac{2}{5}$ , not $\frac{1}{5}$ .
Forgetting the sign at $- \infty$ . $x^{2} = ∣ x ∣$ , which is $- x$ when $x$ is negative. Roots can give a different asymptote at each end.
Treating $\infty - \infty$ as $0$ . It is indeterminate. Rewrite it, usually with the conjugate, before taking the limit.
Thinking the graph never crosses its asymptote. Horizontal asymptotes are about the ends. A graph can cross $y = L$ in the middle, or, like $\frac{s i n x}{x}$ , infinitely often.

Practice it now

Five minutes, no account

Rational limits at infinity come up in a Sprint from about the sixth problem, with equal degrees or a bigger bottom, so the answer is a coefficient ratio or $0$ . In our answer data they are right 92% of the time, so they are where you bank time for the harder ones.

Play a Sprint See it on the formula sheet

FAQ

How do you find a horizontal asymptote?: Take $lim_{x \to \infty} f (x)$ and $lim_{x \to - \infty} f (x)$ . Each finite answer $L$ gives a horizontal asymptote $y = L$ .
What does it mean when the degree of the top is bigger?: The limit is $\infty$ or $- \infty$ and there is no horizontal asymptote. If the top is exactly one degree bigger, there is a slant asymptote instead, found by polynomial division.
Can a function have two horizontal asymptotes?: Yes, one at each end. Rational functions cannot, but functions with roots, like $\frac{4 x ^{2} + 1}{3 x - 5}$ , and exponentials and $arctan x$ often do.
What is the limit of $e^{- x}$ as $x \to \infty$ ?: $0$ , so $y = 0$ is a horizontal asymptote on the right. Exponentials beat every power of $x$ : $lim_{x \to \infty} \frac{x ^{100}}{e ^{x}} = 0$ too, eventually.
Can I use L'Hôpital's rule for limits at infinity?: Yes, when the limit is $\frac{\infty}{\infty}$ or $\frac{0}{0}$ . For rational functions it is slower than the degree rule, but for things like $\frac{x}{e ^{x}}$ it is the quickest way.
Where can I find limits at infinity practice problems?: Right on this page: the practice problems section has three limits at infinity problems, easy, medium and harder, with the answer and the first step under each one, and a box that checks what you type with the same checker that grades Asymptote. The game has 500 more limits at infinity problems after that, free and with no account: try a five-minute Sprint.

Limits at Infinity: Horizontal Asymptotes & Practice Problems

When to use it

Use these rules when

x \to \infty

x \to - \infty

, to find the end behavior of a function or its horizontal asymptotes. For rational functions the degree rules are instant. With square roots, divide by the highest power carefully and watch the sign when

x

is negative. And for a difference of two large things, like

x^{2} + 6 x - x

, multiply by the conjugate first. If you end up with

\frac{\infty}{\infty}

that none of these tidy, L'Hôpital's rule also works at infinity.

How to do it, step by step

Find the highest power of $x$ in the denominator. For

x^{2} + \dots

, that counts as

x^{1}

Divide every term, top and bottom, by that power. This changes nothing about the value, only how it looks.

Send each $\frac{c}{x ^{k}}$ to $0$ . Any constant over a positive power of

x

vanishes as

x \to \pm \infty

Read off what is left. A number is the limit and the horizontal asymptote; anything still growing means the limit is

\infty

- \infty

Check $- \infty$ separately when there are roots. For

x < 0

x^{2} = ∣ x ∣ = - x

, which can flip the sign of the answer.

Worked examples

Example 1: Equal degrees

Easy

Find the limit:

x \to \infty lim \frac{3 x ^{2} - 5 x + 1}{2 x ^{2} + 7}

Top and bottom are both degree 2. Divide every term by $x^{2}$ :

\frac{3 - \frac{5}{x} + \frac{1}{x ^{2}}}{2 + \frac{7}{x ^{2}}} ⟶ \frac{3 - 0 + 0}{2 + 0}

Answer

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

Example 2: A square root on top

Medium

Find the limit, then both horizontal asymptotes:

x \to \infty lim \frac{4 x ^{2} + 1}{3 x - 5}

The top behaves like $4 x^{2} = 2∣ x ∣$ , so it is degree 1, like the bottom. Divide by $x$ . For $x > 0$ , $x = x^{2}$ , so the $x$ can go inside the root as $x^{2}$ :

\frac{4 + \frac{1}{x ^{2}}}{3 - \frac{5}{x}} ⟶ \frac{4}{3} = \frac{2}{3} (x \to \infty)

Answer

x \to \infty lim \frac{4 x ^{2} + 1}{3 x - 5} = \frac{2}{3}

Example 3: Infinity minus infinity

Harder

Find the limit:

x \to \infty lim (x^{2} + 6 x - x)

Both terms grow without bound, and $\infty - \infty$ could be anything. Multiply by the conjugate over itself to turn the difference into a fraction:

(x^{2} + 6 x - x) \cdot \frac{x ^{2} + 6 x + x}{x ^{2} + 6 x + x} = \frac{( x ^{2} + 6 x ) - x ^{2}}{x ^{2} + 6 x + x} = \frac{6 x}{x ^{2} + 6 x + x}

Now divide top and bottom by $x$ (positive, so it goes inside the root as $x^{2}$ ):

\frac{6}{1 + \frac{6}{x} + 1} ⟶ \frac{6}{1 + 1}

So $x^{2} + 6 x$ ends up exactly $3$ more than $x$ , no matter how big $x$ gets. The guess "both are about $x$ , so the answer is $0$ " misses it completely.

Answer

x \to \infty lim (x^{2} + 6 x - x) = 3

Limits at infinity practice problems

Three to try, easy to harder. Type an answer and it is checked instantly by the same checker that grades the game; any correct form counts. Stuck? The answer and the first step are under each one.

Problem 1

Easy

Find the limit:

x \to \infty lim \frac{4 x ^{2} - 3 x + 1}{6 x ^{2} + 5}

Stuck? Show the answer

Answer

\frac{2}{3}

How to start

Equal degrees, so only the leading coefficients matter: divide every term by $x^{2}$ and let the rest vanish.

Problem 2

Medium

Find the limit (note the minus sign):

x \to - \infty lim \frac{4 x ^{2} + 3}{5 x - 2}

Stuck? Show the answer

Answer

- \frac{2}{5}

How to start

For $x < 0$ , $x^{2} = - x$ , so dividing the root by $x$ puts a minus sign in front: $\frac{4 x ^{2} + 3}{x} = - 4 + \frac{3}{x ^{2}}$ .

Problem 3

Harder

Find the limit:

x \to \infty lim (x^{2} + 5 x - x^{2} - 3 x)

Stuck? Show the answer

Answer

4

How to start

Multiply by the conjugate over itself. The top becomes $(x^{2} + 5 x) - (x^{2} - 3 x) = 8 x$ ; then divide top and bottom by $x$ .

Common mistakes

Comparing the wrong coefficients. The rule uses the leading coefficients, the ones on the highest power, not the first numbers written. In

\frac{1 + 2 x ^{2}}{5 x ^{2} - x}

the answer is

\frac{2}{5}

, not

\frac{1}{5}

Forgetting the sign at $- \infty$ .

x^{2} = ∣ x ∣

, which is

- x

when

x

is negative. Roots can give a different asymptote at each end.

Treating $\infty - \infty$ as $0$ . It is indeterminate. Rewrite it, usually with the conjugate, before taking the limit.

Thinking the graph never crosses its asymptote. Horizontal asymptotes are about the ends. A graph can cross

y = L

in the middle, or, like

\frac{s i n x}{x}

, infinitely often.

Practice it now

Five minutes, no account

Play a Sprint See it on the formula sheet

FAQ

How do you find a horizontal asymptote?

Take

lim_{x \to \infty} f (x)

and

lim_{x \to - \infty} f (x)

. Each finite answer

L

gives a horizontal asymptote

y = L

What does it mean when the degree of the top is bigger?

The limit is

\infty

- \infty

and there is no horizontal asymptote. If the top is exactly one degree bigger, there is a slant asymptote instead, found by polynomial division.

Can a function have two horizontal asymptotes?

Yes, one at each end. Rational functions cannot, but functions with roots, like

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

, and exponentials and

arctan x

often do.

What is the limit of

e^{- x}

x \to \infty

0

, so

y = 0

is a horizontal asymptote on the right. Exponentials beat every power of

x

lim_{x \to \infty} \frac{x ^{100}}{e ^{x}} = 0

too, eventually.

Can I use L'Hôpital's rule for limits at infinity?

Yes, when the limit is

\frac{\infty}{\infty}

\frac{0}{0}

. For rational functions it is slower than the degree rule, but for things like

\frac{x}{e ^{x}}

it is the quickest way.

Where can I find limits at infinity practice problems?

Right on this page: the practice problems section has three limits at infinity problems, easy, medium and harder, with the answer and the first step under each one, and a box that checks what you type with the same checker that grades Asymptote. The game has 500 more limits at infinity problems after that, free and with no account: try a five-minute Sprint.

Limits at Infinity: Horizontal Asymptotes & Practice Problems

The formula

When to use it

How to do it, step by step

Worked examples

Example 1: Equal degrees

Example 2: A square root on top

Example 3: Infinity minus infinity

Limits at infinity practice problems

Common mistakes

FAQ

Keep going

Limits at Infinity: Horizontal Asymptotes & Practice Problems

The formula

When to use it

How to do it, step by step

Worked examples

Example 1: Equal degrees

Example 2: A square root on top

Example 3: Infinity minus infinity

Limits at infinity practice problems

Common mistakes

FAQ

Keep going