Limit Calculator
Limit Calculator — Calculate lim f(x) as x→a
This online limit calculator allows you to quickly and accurately calculate the limit of any function at a given point or at infinity. Supports complex functions and shows step-by-step solutions.
What is a function limit?
The limit of a function lim(x→a) f(x) is the value that the function f(x) approaches as the variable x approaches point a. This is a fundamental concept in mathematical analysis.
Types of limits
- Regular limits: lim(x→2) (x² - 4)/(x - 2)
- Limits at infinity: lim(x→∞) sin(x)/x
- One-sided limits: left x→a⁻ and right x→a⁺
- Indeterminate forms: 0/0, ∞/∞, 0·∞, ∞-∞
Applications of limits
- Checking continuity of functions at a point
- Finding asymptotes of function graphs
- Calculating derivatives (definition of derivative)
- Studying behavior of functions near singular points
How to use the calculator?
- Enter the function f(x)
- Specify point a (or ∞ for infinity)
- Choose limit type (two-sided/one-sided)
- Click "Calculate Limit"
Examples: (x^2-1)/(x-1), sin(x)/x, (1+1/x)^x, ln(x), e^x/x^2
Frequently Asked Questions
What is a function limit?
A function limit is the value that a function approaches as the variable approaches a certain point. Denoted as lim(x→a) f(x) = L, where L is the limit value.
When does a limit not exist?
A limit doesn't exist if: left and right limits are not equal, the function oscillates without approaching a specific value, or the result is an indeterminate form that cannot be resolved.
What is L'Hôpital's rule?
L'Hôpital's rule allows calculating limits of type 0/0 or ∞/∞ by finding the limit of the ratio of derivatives: lim f(x)/g(x) = lim f'(x)/g'(x).
How to calculate limit at infinity?
To calculate lim(x→∞) f(x) enter 'infinity' or '∞' in the point field. The calculator will automatically apply appropriate methods.
What indeterminate forms does the calculator support?
Supported main indeterminate forms: 0/0, ∞/∞, 0·∞, ∞-∞, 1^∞, 0^0, ∞^0. The calculator automatically applies needed solution methods.