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# Limit Calculator

Enter a function and the calculator will determine its limits (Negative &amp; Positive, One-tailed &amp; Two-tailed). Get step wise solution for finite and infinite limit simplification with graph.

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Limit calculator evaluates the limit values of a function with respect to the input variable x. Analyze positive and negative limits of any calculus function with either single or multi variables.

Also, the calculator supports $$\frac{0}{0}$$ and $$\frac{\infty}{\infty}$$ limit problems to show you complete steps with visual representation. Just enter the function and see its behaviour at a certain limiting point.

## What Are Limits In Math?

“Limit defines the behavior of a function at a certain point for any input change”

Limit notation represents a mathematical concept that is based on the idea of closeness.

The calculator follows the same technique and assigns values to certain functions at points where no values are defined. It does this all in such a way as to be consistent with proximate or near values.

## Basic Limit Rules:

Limit calculator with steps works by analyzing various limit operations. These laws can be used to assess the limit of a polynomial or rational function manually as well.

Additionally, there are certain conditions for some rules and if they are not satisfied, then the rule cannot be used to validate the evaluation of a limit. Among these rules include:

 Rules Expressions Sum/Difference Rule limx→b[f(x) ± h(x)] = limx→b[f(x)] ± limx→b[h(x)] Power Rule limx→b[f(x)n] = [limx→b[f(x)]]n Product Rule limx→b[f(x) * h(x)] = limx→b[f(x)] * limx→b[h(x)] Constant Rule limx→b[k] = k Quotient Rule limx→b[f(x) / h(x)] = limx→b[f(x)] / limx→b[h(x)] L'Hopital's Rule limx→b[f(x) / h(x)] = limx→b[f'(x) /h'(x)]

## How To Evaluate Limits?

### Example # 01:

Evaluate the limit of the function below:

$$\lim_{x \to 3} 4x^{3}+6x{2}-x+3$$

Solution:

Here we will be using the substitution method:

### Step 01:

Apply a limit to each and every value in the given function separately to simplify the solution:

$$= \lim_{x \to 3} \left(4x^{3}\right)+\lim_{x \to 3} \left(6x^{2}\right) - \lim_{x \to 3} \left(x\right) + \lim_{x \to 3} \left(3\right)$$

### Step 02:

Now write down each coefficient as a multiple of the separate limit functions:

$$= 4 * \lim_{x \to 3} \left(x^{3}\right)+6 * \lim_{x \to 3} \left(x^{2}\right) - \lim_{x \to 3} \left(x\right) + \lim_{x \to 3} \left(3\right)$$

### Step 03:

Substitute the given limit i.e;

$$\lim_{x \to 3}$$:

$$\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 4 * \left(3^{3}\right) + 6 * \left(3^{2}\right) - 3 + 3$$

### Step 04:

Simplify to get the final answer:

$$\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 4 * 27 + 6 * 9 - 3 + 3$$

$$\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 108 + 6 * 9 - 3 + 3$$

$$\lim_{x \to 3} 4x^{3}+6x{2}-x+3 = 162$$

### Example # 02:

$$\lim_{x \to 0} \left(\frac{sin x}{x}\right)$$

#### Solution:

Using The Substitution Method:

$$\lim_{x \to 0} \left(\frac{sin x}{x}\right)$$

$$= \frac{sin 0}{0}$$

$$= \frac{0}{0}$$

Which is an indeterminate form. So here we will be applying l’hopital’s rule: Before we move on, we have to check whether both the functions above and below the vinculum are differentiable or not.

$$\frac{d}{dx} \left(sin x\right) = cos x$$

$$\frac{d}{dx} \left(x\right) = 1$$

Moving ahead further now:

$$\lim_{x \to 0} \left(\frac{cos x}{1}\right)$$

$$= \frac{cos 0}{1}$$

$$= 1$$

## How Does Limit Calculator Function?

The tool is straightforward to use! It requires a few inputs to calculate limits of the given function at any point that include:

Inputs To Enter:

• Enter the function
• Select the variable from the drop-down with respect to which you need to evaluate the limit. It can be x,y,z,a,b,c, or n.
• Specify the number at which you want to calculate the limit. In this field, you can use a simple expression as well such as inf=∞ or pi =π.
• Now select the direction of the limit. It can be either positive or negative
• Tap Calculate

Results You Get:

• Limits of the given function
• Step by step calculations
• Taylor’s series expansion for the given function

## References:

From the authorized source of Wikipedia: Limit (mathematics), function, sequence, standard parts and much more!

The source of Khan Academy provides with: Best Strategy in finding limits

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