What is the limit in Mathematics? Explained with Calculations

In Calculus, limit provide a way to study and analyze the behavior of functions as the input approaches a particular value. The concept of limits allows us to investigate questions such as "What is the value of a function at a specific point?" / "Does a function have a well-defined value as the input gets arbitrarily close to a certain value?"

Evaluating limits involves various techniques, such as direct substitution, factoring, rationalization, and the use of special trigonometric identities. In this article, we will discuss the definition of limit, its rule, and types with solved examples.

What is the limit in Mathematics?

In mathematics, a limit is a fundamental concept that describes the behavior of a function as the input values approach a particular value. It represents the value that a function "approaches" or gets arbitrarily close to, without actually reaching it.

Formally, the limit of a function f(x) as x approaches a value c is denoted as follows:

lim(x → c) f(x) = L

This notation indicates that as x gets arbitrarily close to c (from both sides), the corresponding values of f(x) get arbitrarily close to L. The limit L may exist and be a well-defined value, or it may not exist depending on the behavior of the function.

Rules of Limit

There are several important rules/properties associated with limits that are commonly used in calculus. Here are some of the key rules of limits:




Limit of a Constant

lim(x→a) c =


Limit of the Identity Function

Lim(x→a) x =


Sum and Difference Rule

lim(x→a) [f(x) ± g(x)] =

L ± M

Product Rule

lim(x→a) [f(x) × g(x)] =

L × M

Quotient Rule

lim(x→a) [f(x) / g(x)] =

L / M

Power Rule

lim(x→a) [f(x)n] =    if lim(x→a) f(x) =



lim(x→a) ex =


L'Hôpital's rule is a well-known technique for finding limits involving indeterminate forms, where both the numerator and denominator approach zero or infinity. To deal with this kind of problems, you can get help from an L’hopital’s rule calculator.

Types of limits

Limits can take on different forms based on the behavior of the function as the input approaches the given value. Here are a few common types of limits:

·         Finite Limits: A finite limit exists when the function approaches a specific finite value as the input approaches a particular point.

·         Infinite Limits: An infinite limit occurs when the function grows or diminishes without bounds as the input approaches a certain value. It can approach positive infinity (∞), negative infinity (-∞), or both.

·         Limits at Infinity: Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity. The limit can be a finite value, infinity, or may not exist.

The limit has broad applications across mathematics, science, and engineering. It is used to analyze rates of change, determine critical points, solve optimization problems, define probability distributions, analyze algorithms, and study the behavior of physical systems.

How to Find Limits?

You can take assistance from an online limit calculator to find limits with steps without involving in lengthy calculations. Here are a few examples to find limits manually with the help of rules of limit.

Example 1:

limx→-3(5x3 - 10x2 +15x - 20)

lim x→-3=?


Step 1:

 In the first step, we apply the given limit to the function.

= limx→-3(5x3) - limx→-3(10x2) + limx→-3(15x) - limx→-3(20)

Step 2:

In the second step, we take out the limit function constant value.

= 5limx→-3(x3) -10 limx→-3(x2) +15 limx→-3​(x) - limx→-3​(20)

Step 3:

Apply the value of the limit in the given function. Here, the value of the limit is x=1

Apply the limit value x=1 we get.

limx→-3(5x3 - 10x2+ 15x - 20) = 5(-3)3 -10(-3)2 +15(-3) -20

limx→-3(5x3 - 10x2+ 15x - 20) = -167

Example 2:

limx→-2[​(10x2 +15x - 20)/3x2+1]



we can find the limit of the following function by using step-by-step.

Step 1:

Apply the given limit to the function

limx→-2[​(10x2 +15x - 20)/(3x2+1)] = limx→-2[​(10x2 +15x - 20)/ limx→-2(3x2+1)

step 2:

The coefficient of the function get out of the limit and apply limit to the all function.

limx→-2[​(10x2 +15x - 20)/(3x2+1)] = [10 limx→-2 (x2) +15 limx→-2 (x) -20 limx→-2 (1)]/ [ limx→-2(x2) +1 limx→-2(1)]

step 3:

Now put the limit value

limx→-2[​(10x2 +15x - 20)/(3x2+1)] =10(-2)2+15(-2)-20(1)/ (-2)2+1

step 4:

simplify the question


limx→-2[​(10x2 +15x - 20)/(3x2+1)] =-10/5 =-2


Question 1:

What does the mean of limit existing?


For a limit to exist, the function's output values must approach a specific value as the input values get arbitrarily close to the given value. If this occurs, the limit is said to exist

Question 2:

 How do you evaluate limits algebraically?


To evaluate limits algebraically, you can apply various limit rules and algebraic manipulations. These include the sum, difference, product, quotient, and power rules mentioned earlier. Additionally, factoring, rationalizing, and simplifying expressions can often help in evaluating limits.

Question 3:

How is a limit denoted?


 A limit is denoted using mathematical notation. The general notation for the limit of a function f(x) as x approaches a particular value a is written as:

 lim(x→a) f(x).


In this article, we have discussed the definition of limit, along with its rules, types, and examples. Now you can seek proper guidance about limits from this post. You have to solve several problems by hand to master the calculations of limits.