Limit Calculator

How to Use the Limit Calculator

Enter the function whose limit you want to evaluate, specify the value that x is approaching, and choose the direction. The calculator evaluates the function at points progressively closer to the target and reports the limiting value if the sequence converges.

Classic examples include lim(x→0) sin(x)/x = 1, lim(x→∞) 1/x = 0, and lim(x→0) (e^x−1)/x = 1. You can also examine limits at points where a function has a jump discontinuity by checking left and right limits separately.

Understanding Limits

The concept of a limit is foundational in calculus. A limit describes the behaviour of a function near a point, even when the function is not defined at that point. The formal epsilon-delta definition states that lim(x→a) f(x) = L if for every ε > 0 there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < |x − a| < δ.

One-Sided Limits

A left-sided limit (x→a−) considers only values of x less than a, while a right-sided limit (x→a+) considers only values greater than a. The two-sided limit exists only when both one-sided limits exist and are equal. This distinction matters for functions with jump discontinuities, absolute values, or piecewise definitions.

Limits at Infinity

When evaluating lim(x→∞) f(x), the calculator checks values at increasingly large x to determine horizontal asymptotic behaviour. This is useful for understanding end behaviour of rational functions, exponential decay, and growth rates.

Frequently Asked Questions

What is a limit in calculus?

A limit describes the value a function approaches as x gets closer to a particular value. It is fundamental to defining derivatives, integrals, and continuity.

How does this calculator evaluate limits?

It uses numerical approximation, evaluating the function at points progressively closer to the target from both sides and checking for convergence.

What does it mean when the limit does not exist?

The limit does not exist when left and right limits differ, the function oscillates, or it diverges to infinity near the point.

Can I evaluate limits at infinity?

Yes. Enter “infinity” or “-infinity” as the approach value.

Which functions are supported?

Supports +, −, *, /, ^, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, ln, log10, exp, sqrt, abs, and constants pi and e.