Limit calculator helps you find the limit of a function with respect to a variable. It is an online tool that assists you in calculating the value of a function when an input approaches some specific value.

Limit calculator with steps shows the step-by-step solution of limits along with a plot and series expansion. It employs all limit rules such as sum, product, quotient, and L'hopital's rule to calculate the exact value.

You can evaluate limits with respect to *\(\text{x , y, z , v, u, t}\)* and *\(w\)* using this limits calculator.

That’s not it. By using this tool, you can also find,

- Right-hand limit (+)
- Left-hand limit (-)
- Two-sided limit

To evaluate the limit using this limit solver, follow the below steps.

- Enter the function in the given input box.
- Select the concerning variable.
- Enter the limit value.
- Choose the side of the limit. i.e., left, right, or two-sided.
- Hit the
**Calculate**button for the result. - Use the
**Reset**button to enter new values and the**Keypad icon**to enter additional values.

You will find the answer below the tool. Click on **Show Steps **to see the step-by-step solution.

The limit of a function is the value that *f(x)* gets closer to as *x* approaches some number. Limits can be used to define the derivatives, integrals, and continuity by finding the limit of a given function. It is written as:

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

If *f* is a real-valued function and *a* is a real number, then the above expression is read as,

the limit of *f *of *x *as *x *approaches *a *equals *L.*

Limits can be applied as numbers, constant values (**π, G, k**), infinity, etc. Let’s go through few examples to learn how to calculate limits.

Example - Right-hand Limit

\(\lim _{x\to \:2^+}\frac{\left(x^2+2\right)}{\left(x-1\right)}\)

**Solution:**

A right-hand limit means the limit of a function as it approaches from the right-hand side.

**Step 1: **Apply the limit x➜2 to the above function. Put the limit value in place of **x**.

\(\lim \:_{x\to 2^+}\frac{\left(x^2+2\right)}{\left(x-1\right)}\)

\(=\frac{\left(2^2+2\right)}{\left(2-1\right)}\)

**Step 2: **Solve the equation to reach a result.

\(=\frac{\left(4+2\right)}{\left(2-1\right)} =\frac{6}{1} =6 \)

**Step 3:** Write the expression with its answer.

\(\lim \:_{x\to \:\:2^+}\frac{\left(x^2+2\right)}{\left(x-1\right)}=6\)

Graph

Example - Left-hand Limit

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

**Solution:**

A left-hand limit means the limit of a function as it approaches from the left-hand side.

**Step 1: **Place the limit value in the function.

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

\(=\frac{\left(3^2-3\left(3\right)+4\right)}{\left(5-3\left(3\right)\right)}\)

**Step 2: **Solve the equation further.

\(=\frac{\left(9-9+4\right)}{\left(5-9\right)}\)

\(=\frac{\left(0+4\right)}{\left(-4\right)} =\frac{4}{-4} =-1 \)

**Step 3: **Write down the function as written below.

\(\lim \:_{x\to \:3^-}\left(\frac{x^2-3x+4}{5-3x}\right)=-1\)

Graph

Example - Two-sided Limit

\( \lim _{x\to 5}\left(cos^3\left(x\right)\cdot sin\left(x\right)\right) \)

**Solution:**

A two-sided limit exists if the limit coming from both directions (positive and negative) is the same. It is the same as limit.

**Step 1: **Substitute the value of limit in the function.

\(\lim _{x\to 5}\left(cos^3\left(x\right)\cdot sin\left(x\right)\right)\)

\(=cos^3\left(5\right)\cdot \:sin\left(5\right)\)

**Step 2: **Simplify the equation as we did in previous examples.

\( \lim _{x\to 5}\left(cos^3\left(x\right)\cdot sin\left(x\right)\right) \)

\( =cos^3\left(5\right)\:sin\left(5\right)\)

**Step 3: **The above equation can be considered as the final answer. However, if you want to solve it further, solve the trigonometric values in the equation.

\(=\frac{1141}{50000}\cdot \:-\frac{23973}{25000} =-\frac{10941}{500000} \)

\(\lim \:\:_{x\to \:\:5}\left(cos^3\left(x\right)\cdot \:\:sin\left(x\right)\right)\)

\(=-0.021882 \)

Graph

Does sin x have a limit?

**Sin x** has no limit. It is because, as **x** approaches infinity, the y-value oscillates between 1 and −1.

What is the limit of e to infinity?

The limit of **e** to the infinity (∞) is **e**.

What is the limit as e^x approaches 0?

The limit as e^x approaches 0 is 1.

What is the limit as x approaches the infinity of ln(x)?

The limit as **x** approaches infinity of **ln(x) **is **+****∞**. The limit of this natural log can be proved by reductio ad absurdum.

- If
**x >1ln(x) > 0**, the limit must be positive. - As
**ln(x**. If_{2}) − ln(x_{1}) = ln(x_{2}/x1)**x**, the difference is positive, so_{2}>x_{1}**ln(x)**is always increasing. - If lim
**x→∞ ln(x) = M****∈****R**, we have**ln(x) < M****⇒****x < e**, but^{M}**x→∞**so**M**cannot be in**R**, and the limit must be**+∞**.

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