Limit Calculator: Evaluate Limits With Step-By-Step Methods

Limits are the foundation of calculus. Derivatives, integrals, continuity and asymptotes are all defined using them. This guide explains what a limit means, every method a limit calculator uses, and the special cases where a limit does not exist. Every example below was checked with the SymPy computer algebra system.

Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any calculator mentioned here.

What A Limit Is

A limit describes the value a function gets closer and closer to as the input approaches a certain number. What matters is the trend near the point, not the value at the point itself.

Take f(x) = (x² − 1)/(x − 1). At x = 1, you get 0/0, so the function is undefined there. But try values near 1:

The outputs close in on 2 from both sides. So the limit is 2, even though f(1) does not exist. On a graph, this looks like the line y = x + 1 with a hole at (1, 2).

How To Read Limit Notation

The notation lim (x → a) f(x) = L reads "the limit of f(x) as x approaches a equals L." A few variations show up often:

The formal version, called the epsilon-delta definition, says: for every small distance ε around L, there is a distance δ around a so that every x within δ of a (except a itself) gives f(x) within ε of L. Intro courses rarely ask you to use it, but it is what makes "gets closer and closer" precise.

How To Use The Limit Calculator

The limit calculator above handles limits at a point, at infinity, and in two variables. It takes three steps:

1. Pick a tab. Use Limit at a Point for x → a, Limit at Infinity for x → ∞, or Multivariable Limit for (x, y) → (a, b).
2. Enter the function and the point. Use ^ for powers, sqrt() for roots, and type inf for infinity.
3. Choose a direction if you need a one-sided limit.

The result shows each method used, the left and right limits, a table of values and a graph, so you can see the answer as well as calculate it.

Step 1: Try Direct Substitution

Always start by plugging in the value. If the function is continuous at that point and you get a real number, that number is the limit.

lim (x → 3) (x² + 2x) = 9 + 6 = 15.

Direct substitution works for polynomials everywhere, and for rational, root, exponential, log and trig functions wherever they are defined. If you get a number divided by zero, like 5/0, the limit is infinite or does not exist. If you get 0/0, you have an indeterminate form, and you need one of the methods below.

Indeterminate Forms

An indeterminate form is a result like 0/0 that does not tell you the limit. It means the parts are racing against each other, and you need to simplify before you can tell who wins.

The seven indeterminate forms, as listed in Paul's Online Math Notes at Lamar University, are:

Notice that 0⁰ and 1^∞ do not automatically equal 1. The answer depends on how fast each part moves.

Methods For Solving Limits

When substitution gives 0/0 or another indeterminate form, one of these methods usually works. Pick the method based on what the function looks like.

Factoring And Canceling

Use factoring when the function is a fraction of polynomials. Factor, cancel the shared factor that causes 0/0, then substitute.

lim (x → 3) (x² − 9)/(x − 3):

1. Factor: (x − 3)(x + 3)/(x − 3).
2. Cancel (x − 3): x + 3.
3. Substitute: 3 + 3 = 6.

Canceling is allowed because the limit only cares about x near 3, not at 3, where the two functions differ.

Rationalizing

Use rationalizing when a square root causes the 0/0. Multiply the top and bottom by the conjugate, which flips the sign between the terms.

lim (x → 0) (√(x + 4) − 2)/x:

1. Multiply by (√(x + 4) + 2)/(√(x + 4) + 2).
2. The top becomes (x + 4) − 4 = x.
3. Simplify: x / [x(√(x + 4) + 2)] = 1/(√(x + 4) + 2).
4. Substitute: 1/(2 + 2) = 1/4.

Simplifying Complex Fractions

Use a common denominator when the function has fractions inside fractions.

lim (x → 2) (1/x − 1/2)/(x − 2):

1. Combine the top: 1/x − 1/2 = (2 − x)/(2x).
2. Divide by (x − 2): (2 − x)/[2x(x − 2)] = −1/(2x).
3. Substitute: −1/4.

The key step is noticing that 2 − x = −(x − 2), which lets the factor cancel.

L'HôPital's Rule

Use L'Hôpital's rule for 0/0 or ∞/∞ when algebra gets messy. Take the derivative of the top and the bottom separately, then try the limit again.

lim (x → 0) sin(5x)/x gives 0/0. The derivatives are 5cos(5x) and 1, so the limit is 5cos(0) = 5.

The rule only works on quotients in the forms 0/0 or ∞/∞. For 0 · ∞, rewrite it as a fraction first. For 0⁰, 1^∞ and ∞⁰, take the natural log, find that limit, then raise e to the result. If you need help with the derivatives, the derivative calculator shows each rule.

The Squeeze Theorem

Use the squeeze theorem when a function is trapped between two simpler functions that have the same limit. If g(x) ≤ f(x) ≤ h(x) near a, and g and h both approach L, then f must approach L too.

lim (x → 0) x²·sin(1/x):

1. sin(1/x) is always between −1 and 1.
2. So −x² ≤ x²·sin(1/x) ≤ x².
3. Both −x² and x² approach 0.
4. So the limit is 0.

This works even though sin(1/x) itself has no limit at 0. The x² factor crushes the wiggles. The same idea proves lim (x → ∞) sin(x)/x = 0, since the top stays between −1 and 1 while the bottom grows without bound.

Special Trig Limits

Two trig limits come up so often that it helps to memorize them:

• lim (x → 0) sin(x)/x = 1
• lim (x → 0) (1 − cos x)/x = 0

They only work with x in radians. In degrees, the first one would be π/180 instead of 1. You can see the first limit in a values table: sin(0.1)/0.1 ≈ 0.998334, sin(0.01)/0.01 ≈ 0.999983, getting closer to 1.

One-Sided Limits And When A Limit Does Not Exist

A two-sided limit exists only if the left-hand and right-hand limits both exist and are equal. If they differ, or if the function never settles down, the limit does not exist.

The function |x|/x equals −1 for every negative x and 1 for every positive x. The graph jumps at 0, so there is no single value to approach.

Saying a limit "equals ∞" is a shorthand. It means the function grows without bound, and technically the limit does not exist as a real number. Many teachers still want you to write ∞ to show how it fails.

Limits Of Piecewise Functions

A piecewise function uses different formulas on different intervals. At the point where the formula switches, find each one-sided limit with the formula for that side, then compare.

For f(x) = x + 1 when x < 2, and 3x − 3 when x ≥ 2:

• Left of 2, use x + 1: the limit is 2 + 1 = 3.
• Right of 2, use 3x − 3: the limit is 6 − 3 = 3.
• Both sides agree, so lim (x → 2) f(x) = 3.

Now change it to g(x) = 2x when x < 1, and x + 3 when x ≥ 1. The left limit at 1 is 2 and the right limit is 4. They differ, so the limit at x = 1 does not exist, even though g(1) = 4 is defined.

Limits At Infinity

A limit at infinity shows what a function does as x grows very large. For rational functions, compare the highest power of x on the top and on the bottom.

A quick way to see why: divide every term by the highest power in the denominator. Terms like 1/x and 1/x² shrink to 0, leaving only the leading terms. This follows the method shown in Lamar University's limits at infinity notes.

When the limit at infinity is a number L, the line y = L is a horizontal asymptote. So y = 3/5 is a horizontal asymptote of (3x² + 1)/(5x² − x).

For other functions, think about growth rates. Exponentials beat powers, and powers beat logs. So eˣ/x² → ∞ and ln(x)/x → 0 as x → ∞.

One famous limit at infinity defines the number e: lim (x → ∞) (1 + 1/x)ˣ = e ≈ 2.71828. This is where continuous compound interest comes from.

Infinite Limits And Vertical Asymptotes

An infinite limit happens when the function grows without bound as x approaches a finite number. These points are vertical asymptotes on the graph.

lim (x → 2) 1/(x − 2)² = ∞, because the bottom gets tiny and stays positive on both sides. The line x = 2 is a vertical asymptote.

To find vertical asymptotes of a rational function, look for x-values that make the denominator zero after you cancel common factors. If a factor cancels, you get a hole instead of an asymptote. That is the difference between (x² − 1)/(x − 1), which has a hole at x = 1, and 1/(x − 1), which has an asymptote there.

Limit Laws

Limit laws let you break a complicated limit into simple pieces, as long as each piece has a limit.

The quotient law breaks down when the bottom's limit is 0. That is exactly when you need the methods above.

Limits And Continuity

A function is continuous at x = a when three things are true: f(a) exists, the limit as x → a exists, and the two are equal. In plain terms, you can draw the graph through that point without lifting your pencil.

A removable discontinuity can be "fixed" by defining the function at the hole. If you set f(1) = 2 for (x² − 1)/(x − 1), it becomes continuous everywhere.

How Limits Define Derivatives

The derivative is itself a limit. It is the limit of the slope between two points as they slide together: f′(x) = lim (h → 0) [f(x + h) − f(x)]/h. Every derivative rule is proven from this limit.

Definite integrals are limits too. They are the limit of a sum of thinner and thinner rectangles under a curve. So the integral calculator and the derivative calculator both rest on the same idea. For all three topics in one place, try the calculus solver.

Multivariable Limits

A limit in two variables, like (x, y) → (0, 0), must give the same value along every possible path to the point. If two paths give different answers, the limit does not exist.

For f(x, y) = xy/(x² + y²) at (0, 0):

• Along y = 0: f = 0/x² = 0, so the limit is 0.
• Along y = x: f = x²/(2x²) = 1/2, so the limit is 1/2.

The paths disagree, so the limit does not exist.

Proving a two-variable limit does exist is harder, because there are infinitely many paths. The usual tools are the squeeze theorem or switching to polar coordinates, where x = r·cos(θ) and y = r·sin(θ), and checking whether the result goes to the same value as r → 0 for every angle.

Estimating A Limit With A Table Or Graph

When you cannot simplify a limit, you can estimate it numerically. Plug in x-values closer and closer to the point from both sides and watch where the outputs head.

This method has limits of its own. Rounding can fool you when the values get very small, and a function can look like it settles before it suddenly changes. Use a table to check an answer, not to prove one.

On a TI-84, there is no built-in limit command. Enter the function in Y=, open TABLE SETUP, set the table to start just below the point with a tiny step, and read the values as x approaches from each side. A graph with a zoom-in at the point also helps you spot holes and jumps. The scientific calculator can evaluate the function at individual points.

Common Limit Mistakes

Most limit errors come from stopping at 0/0 or misreading what the notation asks for.

Summary

• A limit is the value a function approaches near a point, not the value at the point.
• Try substitution first. A number means you are done, and 0/0 means you need to simplify.
• Use factoring, rationalizing, common denominators, L'Hôpital's rule or the squeeze theorem for indeterminate forms.
• A two-sided limit exists only when the left and right limits match.
• At infinity, compare degrees for rational functions. A finite limit gives a horizontal asymptote.
• Two-variable limits must match along every path.

Next step: enter your limit into the limit calculator at the top of this page. For more math tools, visit the Calc Solver homepage.

Frequently Asked Questions