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 the left-hand limit
Evaluate \(\lim _{x\to 3^-}\left(\frac{5x^3+x-3}{3-x^2}\right)\)
Solution
A function that approaches from the left-hand side is known as the 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 the 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.
\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(5\left(2\right)^2-3\right)+\left(3\left(2\right)-4\right)\)
\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(5\left(4\right)-3\right)+\left(3\left(2\right)-4\right)\)
\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(20-3\right)+\left(6-4\right)\)
\( \lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=\left(17\right)+\left(2\right)\)
\(\lim _{x\to 2}\left[\left(5x^2-3\right)+\left(3x-4\right)\right]=19\)