Limit Calculator

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Limit calculator with steps

Limit calculator is used to find the limit of the function at any point w.r.t a variable. This limit solver evaluates the left-hand, right-hand, and two-sided limits. It calculates the limit with a step-by-step solution.

How does the limits calculator work?

Follow the below steps to find the limits of the functions.

  • Enter the function into the input box.
  • Use the keypad icon to enter math symbols.
  • Select the variable.
  • Select the side of the limit i.e., left hand, right hand, or two-sided.
  • Write the limit value.
  • If you want sample examples, click the load example
  • Press the calculate button to get the result.
  • To enter a new function, click the clear

What are the limits?

In mathematics, a limit is an amount that a function approaches as the input approaches some value. Limits are important in calculus and mathematical analysis. It is also used to define derivatives, integrals, and continuity.

The equation used to represent limits is given below.

\(\lim _{x\to c}f\left(x\right)=L\)

This equation can be read as the limit of f of x as x approaches c equals L. If the function makes \(\frac{0}{0}\) or \(\frac{\infty }{\infty }\) form then L’hospital’s rule is applied on the function to evaluate the limits.

Types of limits

There are three types of limits.

  • Left-hand limit
  • Right-hand limit
  • Two-sided limits

Rules of limit

Following are some rules of limits.

Names

Rules

Power rule

\(\lim _{x\to c\:}\left(f\left(x\right)\right)^k=\left(\lim _{x\to c\:}f\left(x\right)\right)^n\)

Sum rule

\(\lim _{x\to c\:}\left(f\left(x\right)+g\left(x\right)\right)=\lim_{x\to c\:}f\left(x\right)+\lim _{x\to c\:}g\left(x\right)\)

Difference rule

\(\lim _{x\to c}\left(f\left(x\right)-g\left(x\right)\right)=\lim _{x\to c}f\left(x\right)-\lim _{x\to c}g\left(x\right)\)

Product rule

\(\lim _{x\to c}\left(f\left(x\right)\cdot g\left(x\right)\right)=\lim _{x\to c}f\left(x\right)\cdot \lim _{x\to c}g\left(x\right)\)

Quotient rule

\(\lim _{x\to c}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{\lim _{x\to c}f\left(x\right)}{\lim _{x\to c}g\left(x\right)}\)

How to calculate limits?

Following are some examples of limits solved by our limit calculator.

Example 1: For left-hand limit

Evaluate \(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)\)

Solution

A function that approaches form left hand side is known as left hand limit of that function.

Step 1: Apply the quotient rule of limit.

\(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)=\frac{\lim _{x\to 3^-}\left(5x^3+x-3\right)}{\lim _{x\to 3^-}\left(3-x^2\right)}\)

Step 2: Now apply the limit and solve the equation.

\(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)=\frac{\left(5\left(3\right)^3+\left(3\right)-3\right)}{\left(3-\left(3\right)^2\right)}\)

\(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)=\frac{\left(5\left(27\right)+3-3\right)}{\left(3-\left(9\right)\right)}\)

\(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)=\frac{\left(135+0\right)}{\left(-3\right)}\)

\(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)=-\frac{135}{3}\)

\(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)=-45\)

Example 2: For right-hand limit

Evaluate \(\lim _{x\to 4^+}\left[\left(15x^2+x^3-5\right)\cdot \left(2x-x^2\right)\right]\)

Solution

Step 1: Apply the product rule of limit.

\( \lim _{x\to 4^+}\left[\left(15x^2+x^3-5\right)\cdot \left(2x-x^2\right)\right]=\lim _{x\to 4^+}\left(15x^2+x^3-5\right)\cdot \lim _{x\to 4^+}\left(2x-x^2\right)\)

Step 2: Now apply the limit and solve the equation.

\(\lim \:_{x\to \:4^+}\left[\left(15x^2+x^3-5\right)\cdot \:\left(2x-x^2\right)\right]=\left(15\left(4\right)^2+\left(4\right)^3-5\right)\cdot \:\left(2\left(4\right)-\left(4\right)^2\right)\)

\( \lim _{x\to 4^+}\left[\left(15x^2+x^3-5\right)\cdot \left(2x-x^2\right)\right]=\left(15\left(16\right)+\left(64\right)-5\right)\cdot \left(8-\left(16\right)\right)\)

\( \lim _{x\to 4^+}\left[\left(15x^2+x^3-5\right)\cdot \left(2x-x^2\right)\right]=\left(240+59\right)\cdot \left(-8\right)\)

\( \lim _{x\to 4^+}\left[\left(15x^2+x^3-5\right)\cdot \left(2x-x^2\right)\right]=\left(299\right)\cdot \left(-8\right)\)

\( \lim _{x\to 4^+}\left[\left(15x^2+x^3-5\right)\cdot \left(2x-x^2\right)\right]=-2392\)

Example 3: For two-sided limit

Evaluate \( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]\)

Solution

Step 1: Apply the sum rule of limit.

\(lim_{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\lim _{x\to 2}\left(5x^2-3\right)+\lim _{x\to 2}\left(3x-4\right)\)

Step 2: Now apply the limit and solve the equation.

\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(5\left(2\right)^2-3\right)+\left(3\left(2\right)-4\right)\)

\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(5\left(4\right)-3\right)+\left(3\left(2\right)-4\right)\)

\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(20-3\right)+\left(6-4\right)\)

\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(17\right)+\left(2\right)\)

\(\lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=19\)

Reference

Limit (mathematics): Definition and formula |Wikipedia

Rules of limits. (n.d.)

 How to calculate limits? Calculus I - computing limits. (n.d.)

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