Limit Calculator

Calculate limits step by step

This free calculator will try to find the limit (two-sided or one-sided, including left and right) of the given function at the given point (including infinity), with steps shown.

Enter a function:

Choose a variable:

Find the limit at:

If you need `oo`, type inf.

Choose a direction:

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Limit Calculator is an online tool that finds the limit of a given function by displaying each step of the process. It calculates the limit for a particular variable and gives you the option to choose the limit type: two-sided, left-handed, or right-handed.

How to Use the Limit Calculator?

  • Input

    Start by entering the function for which you want to find the limit into the specified field. Specify the variable (if the function has more than one variable). Specify the value to which the variable is approaching. This can be a numeric value, positive infinity, or negative infinity. Select the type of limit: two-sided, left-handed, or right-handed.

  • Calculation

    Click the "Calculate" button.

  • Result

    The calculator will then display the limit value.

What Are Limits in Math?

In mathematics, a limit is the value $$$L$$$ of the function $$$f(x)$$$ when the input (or variable) $$$x$$$ approaches a certain value, let's call it $$$c$$$. It determines the behavior of the function near the point $$$x=c$$$, but not at this point itself (where the function can be undefined or $$$x$$$ can approach positive or negative infinity).

Definition of a Limit

The limit of a function $$$f(x)$$$ as $$$x$$$ approaches $$$c$$$ is $$$L$$$ if, for every tiny positive number $$$\epsilon$$$, there exists a number $$$\delta$$$ such that if the absolute difference between $$$x$$$ and $$$c$$$ is less than $$$\delta$$$ (but not zero), then the absolute difference between $$$f(x)$$$ and $$$L$$$ is less than $$$\epsilon$$$.

In other words, $$$\lim_{x\to c}f(x)=L$$$ if for every $$$\epsilon\gt0$$$, there exists $$$\delta\gt0$$$ such that $$$0\lt\lvert x-c\rvert\lt\delta$$$ implies $$$\left\lvert f(x)-L\right\rvert\lt\epsilon$$$.



This means that, as $$$x$$$ gets closer to $$$2$$$, $$$x^2+3x+2$$$ approaches $$$12$$$.

Special Types of Limits

  • Infinite Limits. $$$f(x)$$$ approaches infinity (positive or negative) as $$$x$$$ approaches some value. This can be written as

    $$\lim_{x\to c}f(x)=\infty$$

    For instance,

    $$\lim_{x\to 0^+}\frac{1}{x}=\infty$$
  • Limits at Infinity. $$$f(x)$$$ approaches a specific finite value as $$$x$$$ approaches positive or negative infinity:


    For example,


    This means that, as $$$x$$$ gets larger and larger, the value of $$$\frac{1}{x}$$$ gets closer and closer to $$$0$$$.

  • One-Sided Limits. One-sided limits examine the behavior of a function as $$$x$$$ approaches some value from only one side:

    • From the Left (left-hand limit). Notation: $$$\lim_{x\to c^-}f(x)=L$$$. For example, $$$\lim_{x\to0^-}\frac{1}{x}=-\infty$$$.
    • From the Right (right-hand limit). Notation: $$$\lim_{x\to c^+}f(x)=M$$$. For example, $$$\lim_{x\to0^+}\frac{1}{x}=\infty$$$.

    If the one-sided limits don't equal each other, the two-sided limit doesn't exist; if they do, then the two-sided limit also equals that value. For example, $$$\lim_{x\to0}\frac{1}{x}$$$ doesn't exist since $$$\lim_{x\to0^-}\frac{1}{x}\ne\lim_{x\to0^+}\frac{1}{x}$$$.

In calculus, the concept of limit is fundamental. It provides the basis for many other concepts used in the study of functions and their behavior. It offers a way to study a function at a point whose value cannot be calculated directly by examining its behavior as we get as close to that value as we like.

Why Choose Our Limit Calculator?

  • Accuracy

    Our limit calculator uses advanced algorithms, ensuring you get accurate and correct results every time you use it.

  • Explanations

    Our calculator will not only give you the final answer but will carefully guide you through each step of the solution. This detailed approach is especially useful for students and teachers who want to understand the process.

  • User-Friendly Interface

    Our calculator has an intuitive design that makes it easy for even calculus beginners to navigate and use this tool.


What is the primary function of the Limit Calculator?

The Limit Calculator is designed to evaluate the limit of a function as its variable approaches a specific value, providing both the result and the step-by-step solution.

Can the calculator handle one-sided limits?

Absolutely! Our calculator can find both left-hand and right-hand limits, which provides insight into the behavior of the function.

How accurate is the Limit Calculator?

Our calculator uses advanced mathematical algorithms, ensuring correct and accurate results.

Are there any restrictions on the type of functions I can input?

Our Limit Calculator handles a wide range of functions, including polynomial, exponential, logarithmic, and trigonometric functions.