Limits are used extensively in calculus and that’s why holds utmost importance in mathematics. A limit describes the action of a function near a threshold and operates the behavior of functions.

Most of the time, we plug in the value of limit in the function to know its behavior. Because of the complexities involved, sometimes, it becomes tricky to evaluate a function’s value by just applying the limit

Even though an online *limit calculator* could be very handy to deal with those complexities, we will explain basic limit rules and laws for a better understanding.

Let ** f(x)** be defined for all

$$\lim_{x \to a} f(x) = L$$

If, for every ε > 0, there exists a δ > 0, such that if \( 0 < |x−a| < δ, then |f(x) − L| < ε\).

Limits can be implemented to derivatives, integrals, sum, sequences, etc.

A graphical representation of limit for the function ** f (x) = 1/x** for

Rules of limits could be applied according to the algebraic operations such as sum, difference, root, etc.

To elaborate these laws, we will assume ** L, M, k,** and

Moreover, for practical application, we will suppose that the function ** f(x) **approaches a limit value of

Sum rule of limits adds both functions and their limit values.

$$\lim_{x \to a} (f(x) + g(x)) = L + M$$

Let’s substitute limit values for both functions to see the effect of sum rule.

$$\lim_{x \to a} (f(x) + g(x)) = 5 + 8 = 13$$

The difference rule should be applied when there is an expression containing a subtraction operation.

$$\lim_{x \to a} (f(x) - g(x)) = L - M$$

$$\lim_{x \to a} (f(x) - g(x)) = 5 - 8 = -3$$

For product, multiply the limit values to get the product of two functions approaching specifics limits.

$$\lim_{x \to a} (f(x) * g(x)) = L * M$$

$$\lim_{x \to a} (f(x) * g(x)) = 5 * 8 = 40$$

If there is a constant multiplying with the function ** f(x), **it will also be multiplied with the limit value.

$$\lim_{x \to a} (k * f(x)) = k * L$$

Suppose the constant in this case is *3.*

$$\lim_{x \to a} (3 * f(x)) = 3 * 5 = 15$$

The power rule applies in the case of an exponential function.

$$\lim_{x \to a} (f(x))^n = L^n$$

** n** is a positive integer

Suppose *n = 4.*

$$\lim_{x \to a} (f(x))^n = 5^4 = 625$$

Limits can be applied on roots where ** n **will be a positive number.

$$\lim_{x \to a} ^n\sqrt{f(x)} = ^n\sqrt{L} = L^{\frac{1}{n}}$$

** n** is a positive integer

Suppose *n = 4.*

$$\lim_{x \to a} ^4\sqrt{f(x)} = ^4\sqrt{5} = 5^{\frac{1}{4}}$$

When dealing with quotients, we will divide the limits of both functions.

$$\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{L}{M}$$

$$\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{5}{8} = 0.625$$

Now that you know all of the above rules, let’s discuss L'Hopital's Rule which is significant considering the wide applications of limit calculator.

L'Hopital was a French mathematician who proposed that,

**“**When one function is divided by another function, the limit would be same if we take the derivative of each function.**”**

$$\lim_{x \to a} f(x) / g(x) = \lim_{x \to a} f'(x) / g'(x)$$

L'Hopital's Rule is a method to solve indeterminate or unspecified limits.

**References**

Michael Spivak, Calculus, 4th edition, Published, 2008.

Peter Brown, Limits and continuity, A guide for teachers.