What is Limit in Calculus? - Definition, Types & Properties

Introduction

We say that a function has a limit L when the variable x is approached to a point in the domain of the function. This guide will help you to understand the concept of limit and its types.

Limit of a Function

The concept of limit is a fundamental concept of calculus which deals with the change in a function. The limit of a function is a value in its domain that the function approaches as its argument approaches to a.

Let f(x) be a function and there is a point ‘a’ in the domain of f such that when x approaches to a, there will be a number L in the range of f is called the limit of f.

In notation form, we write the limit of f(x) as:

Which is read as the limit of f(x) as x approaches to a is L.

Understanding of Limits

The limit of a function plays an important role in calculus and mathematical analysis. We say that the limit of a function exists if it approaches some values in the range when x approaches to a.

Limits are used in derivatives, integrals and continuity. It is also used to analyse the local behaviour of a function near a point of interest.

6 limit properties and rules

The properties of limits are based on the properties of algebraic operations. Suppose that f(x) = M and g(x) = N The limit of any function can be expressed by using following properties.

1. Sum
$$f(x)\;+\;g(x)\;=\;M\;+\;N$$
2. Difference
$$f(x)\;-\;g(x)\;=\;M\;-\;N$$
3. Product
$$(\;f(x)\;)(\;g(x)\;)\;=\;M\;N$$
4. Division
$$\frac{f(x)}{g(x)}\;=\;\frac{M}{N} $$
5. Power
$$f(x)^k\;=\;M^k \;\;for\;k\;>\;0$$
6. Polynomial

If f(x) and g(x) are polynomials then suppose g(a) ≠ 0, then

The result of limits by these properties is only true if the limit of both functions exists. If the limit of one of them does not exist then their sum, product, difference or division might not exist. It might be approaching infinity.

Types of Limits of Functions

There are three major types of limits. These are

1. One-sided limit
2. Two-sided limit
3. Infinite limit

1. One-sided Limit

A one-sided limit considers only one-sided values of a function. These values can be either positive or negative. The positive values refer to the right-sided limit and the negative values refer to the left-sided limit.

A function f has right-sided limit if it approaches to a such that

$$f(x)\;=\;L$$

And the left-sided limit is,

$$f(x)\;=\;L$$

Where,

• x → a^- shows that the function takes the negative values.
• x → a⁺ shows that the function takes positive values only.
2. Two-sided Limit

A two-sided limit contains both left and right-sided values of a function. By definition of two-sided limit, if one-sided limits left or right are same then,

$$f(x)\;=\;L$$
3. Infinite Limit

A function has an infinite limit when x get closer to a but the value of f(x) gets bigger and bigger, it increases without bound. So, it has a large positive number.

Similarly if the limit has a large negative number as x get closer to a but f(x) decreases without bound then,

Limit at Infinity

In general we write the limit of a function at infinity,

It is defined as the function gets close to L by taking x sufficiently large or it approaches infinity. If this limit exists, we say that the f has the limit L if x is increasing without bound. Similarly,

If this limit exists, it means that f has the limit M if x is decreasing without bound.

Some important formulas of Limits

There are some important rules of limits that are mostly used to solve the limit of a function. These are

1. For all real values of n,
$$\lim_{x→a}\frac{x^n\;-\;a^n}{x\;-\;a}\;=\;na^{(n-1)}$$
2. $$\lim_{∅→0}\frac{sin∅}{∅}\;=\;1$$
3. $$\lim_{∅→0}\frac{tan∅}{∅}\;=\;1$$
4. $$\lim_{∅→0}\frac{∅}{∅}\;=\;0$$
5. $$\lim_{∅→0}cos∅ \;=\;1$$
6. $$\lim_{x→0}e^x\;=\;1$$
7. $$\lim_{x→0}\frac{e^x\;-\;1}{x}\;=\;1$$
8. $$\lim_{x→0}(1\;+\;\frac{1}{x})^x\;=\;e$$

How to Solve the Limit of a Function?

The limit of a function can be calculated by using the limit formula.You can also calculate the limit with the help of limit calculator. Let’s see an example to understand solve the limit.

Example

For the following limit define a, f(x) and L. Also find the limit of a function.

Solution

For a function f, when x approaches to a then,

Now to solve limit, we will perform the following steps,

Since,

So,

So,

Here,

Does a function need to be continuous to have a limit?

A function is said to be continuous if

• The limit of the function is equal to the value of the function at that point.
$$f(x)\;=\;f(a)$$
• The limit of function should exist and left and right-sides limits should be equal.
$$f(x)\;=\;f(x)$$

There may be conditions where one of the above is true and other is not. So, a discontinuous function can have limit but either

Its limit value is not equal to the function value.

Or,

One of the right or left-sided limits becomes undefined. i.e. if f(x) exists then f(x) does not exist.

FAQ’s

Can the Limit of a Function be Zero?

Yes, the limit of a function can be zero. For a function f(x), when x gets closer and closer to zero, the f(x) will get closer to zero. It means that when x approaches infinity, the limit of f(x) will become zero. In notation form,

Can a Limit of a Function be Negative?

When a function has a one-sided limit from the left side, where it takes the negative values. It means that the function has a negative limit. We can express the negative limit for a function f as,

Can the Limit of a Function be Infinity?

Yes, the limit of a function can be infinite. It can lead to positive or negative infinity. A function has an infinite limit when x get closer to a but the value of f(x) gets bigger and bigger, it increases without bound. So, it has a large positive number.

Similarly if the limit has a large negative number as x get closer to a but f(x) decreases without bound then,

Can a Function have Multiple Limits?

A function can have only a limit from the right and a different limit from the left side. But it cannot have more than one limit applied on it. It can just have more than one value on the limit.