Free Calculus Calculator

Limit Calculator

Calculate mathematical limits as x approaches a specified value. Evaluate two-sided and one-sided limits for algebraic, polynomial, rational, trigonometric, logarithmic, and exponential expressions.

Calculate a Limit

Enter your expression using x, then enter the value that x approaches.

Supported syntax: Use x for the variable, ^ for powers, and standard functions such as sin(x), cos(x), tan(x), sqrt(x), ln(x), log(x), exp(x), and abs(x).
x^2 (x^2-1)/(x-1) sin(x)/x (1-cos(x))/x^2 ln(1+x)/x
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What Is a Limit?

A limit describes the value that a function approaches as its input gets closer to a particular value.

The standard notation is:

limx → a f(x)

This asks what value f(x) approaches as x gets closer and closer to a.

The function does not necessarily have to be defined at x = a for the limit to exist.

One-Sided Limits

A left-hand limit approaches a point from values smaller than the target:

limx → a⁻ f(x)

A right-hand limit approaches from values greater than the target:

limx → a⁺ f(x)

A two-sided limit exists when the left-hand and right-hand limits agree.

Common Limits in Calculus

Several standard limits are especially important in calculus.

limx → 0 sin(x) / x = 1
limx → 0 (1 − cos(x)) / x² = 1/2
limx → 0 (eˣ − 1) / x = 1
limx → 0 ln(1 + x) / x = 1

These standard limits are frequently used when simplifying more complicated calculus problems.

How to Calculate a Limit

The appropriate method depends on the expression. Common techniques include:

  • Direct substitution: substitute the limiting value when the function is continuous.
  • Factoring: factor algebraic expressions to simplify removable discontinuities.
  • Rationalization: use a conjugate to simplify expressions containing radicals.
  • Standard limits: apply known trigonometric, exponential, and logarithmic limits.
  • One-sided limits: separately examine behavior from the left and right.
  • L'Hôpital's Rule: differentiate numerator and denominator for suitable indeterminate forms.

Example: A Removable Discontinuity

Consider:

limx → 1 (x² − 1)/(x − 1)

Direct substitution gives the indeterminate form 0/0. Factor the numerator:

x² − 1 = (x − 1)(x + 1)

Therefore:

(x² − 1)/(x − 1) = x + 1

The limit is then:

limx → 1 (x + 1) = 2

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Frequently Asked Questions

What is a limit in calculus?

A limit describes the value that a function approaches as its input gets closer to a specified value.

What is a one-sided limit?

A one-sided limit examines the behavior of a function from only one direction. A left-hand limit approaches from smaller x-values, while a right-hand limit approaches from larger x-values.

Does a function need to be defined at the limit point?

No. A limit can exist even when the function is undefined at the point being approached. For example, removable discontinuities can have well-defined limits.

What happens when the left and right limits differ?

If the left-hand and right-hand limits approach different values, the two-sided limit does not exist.

What does a limit of infinity mean?

If the function grows without bound as x approaches a particular value, the limit can be described as +∞ or −∞. Infinity is not a finite real-number result; it describes unbounded behavior.

Can every limit be calculated numerically?

Numerical sampling works well for many common limits, but some highly oscillatory or complicated expressions require symbolic techniques such as algebraic simplification, standard limit identities, or L'Hôpital's Rule.