In calculus, a limit is defined as a value that a function (or sequence) approaches as the input (or index)

approaches a particular value. It helps us to understand the behavior of functions and sequences near a

specific point or at infinity. Limits are considered the foundation of continuity, derivative, and series.

In this article, we will read the definition, types, and properties of limits. In addition, we will learn how to

calculate limits problems. Further, to gain a better understanding, we will solve different examples in the

example section.

## Definition of Limit Calculus

Consider a function f(x) defined on an interval that contains the point c. We say that the limit of f(x) as x

approaches c is L, denoted by Lim x → c f(x) = L if, for any given positive value epsilon (ε > 0), there exists a

positive value delta (δ > 0) such that |f(x) – L| < ε for all x in the interval (x – δ, x + δ) that contain c.

**Example:**

Consider a function f(x) = 3x – 2. Find out the limit of the given function as x approaches 2.

**Solution:**

By using the definition of limit, we want to find L such that | (3x – 2) – L| < ε for all x satisfying 0 < | x – 2| < δ.

Substitute x = 2 into f(x), we get f(2) = 3(2) – 2 = 4

To ensure | (3x – 2) – 4| < ε, we choose δ = ε / 3.

Hence, the limit of f(x) as x approaches 2 is 4, and for any positive ε, we can find a corresponding δ = ε / 3 such

that | (3x – 2) – 4| < ε for all x satisfying 0 < | x – 2| < δ.

## Properties of Limits

While dealing with limits, the following properties can help in simplifying calculations.

If Lim x→ a f(x), and Lim x→ a g(x) exist, and k is any constant then,

1. Lim x→ a [k f(x)] = k lim x→ a [f(x)]

2. Lim x→ a [f(x) ± g(x)] = Lim x→ a f(x) ± Lim x→ a g(x)

3. Lim x→ a [f(x). g(x)] = Lim x→ a f(x)). Lim x→ a g(x)

4. Lim x→ a (f(x) / g(x)) = Lim x→ a f(x) / Lim x→ a g(x) (provided lim x→ a g(x) ≠ 0)

5. Lim x→ a k = k

6. Lim x→ a (f(x)) k = (Lim x→ a f(x)) k

## Types of Limit

There are several types of limits, but some common ones are given below:

**One-sided limits**are limits of a function as its input approaches a specific point from one side only

(either left or right side). The left-hand limit is expressed as lim x → c – f (x), and the right-side limit is

written as lim x → c + f (x).-
**Two-sided limits**are the most common type of limit. They describe the behavior of function from both

the left and right sides. -
**Limit at Infinity**refers to the limits of a function when its input approaches positive or negative infinity,

expressed as lim x →± ∞ f(x) -
**An infinite limit**occurs when the limit of a function tends to positive or negative infinity as the input

approaches a certain point, denoted as lim x → c f(x) = ± ∞. - Discontinuous limits happen when the limit of a function does not exist at a specific point due to a

break in the graph of the function.

## Methods for Calculating the Limit

Here are some methods and techniques for calculating the limit of the function

### 1. Direct Substitution Method

When the function is defined at the value where the limit is being determined, then put this value into

the given function and calculate the result.

### 2. Factoring and canceling

When the function has an algebraic expression, you can try to find the factor and cancel the common

factors to simplify the given expression before calculating the limit.

### 3. Rationalization

When the function involves radical or complex fractions, you can use the rationalization method to

eliminate radicals and simplify the fraction.

### 4. Trigonometric Identities

You can use the following trigonometric identities if the function contains trigonometric functions.

- lim x→ 0 (1 – cos x / x) = 1
- lim x→ 0 (Sin x / x) = 1
- lim x→ 0 (tan x / x) = 1

### 5. Exponential and logarithmic limits

You can use the following formulas if the function contains exponential functions.

- lim x→ 0 (1 + x) (1/x) = e
- lim x→ ∞ (e x ) = ∞
- lim x→ – ∞ (e x ) = 0

### 6. L Hopital’s Rules

It evaluates limits of indeterminate forms (like 0/0, or ∞/∞). It involves taking the derivative of the

numerator and denominator individually and then evaluating the limit again.

## Solved Examples of Limit

Here are some examples of finding limits with the help of limits rules manually.

**Example 1.**

Evaluate lim x→ 2 [(x 3 + 5x 2 + x + 5) / (x + 5)]

**Solution**

Lim x→ 2 [(x 3 + 5x 2 + x + 5) / (x + 5)]

= Lim x→ 2 [(x 2 (x + 5) + 1(x + 5)) / (x + 5)]

= Lim x→ 2 [(x 2 + 1) (x + 5)) / (x + 5)]

After canceling the common factor, we have

= Lim x→ 2 (x 2 + 1)

After applying the limit, we get

= 2 2 + 1 = 4 + 1 = 5

Hence, lim x→ 2 [(x 3 + 5x 2 + x + 5) / (x + 5)] = 5

**Example 2.**

**Determine the lim x → ∞ (x 2 + 3) / (x 3 + 2)**

**Solution:**

Lim x → ∞ (x 2 + 3) / (x 3 + 2) = ∞ / ∞

This is an indefinite form. Therefore, we will solve this problem with L Hopital’s Rules.

After taking the derivative of the numerator and denominator individually. We get

= Lim x → ∞ (2x / 3x 2 )

After cancellation, we have

= Lim x → ∞ (2 / 3x)

Now apply the limit

= 2 / 3(∞) = 2 / ∞ = 0 (∴1/ ∞ = 0)

Hence, the Lim x → ∞ (x 2 + 3) / (x 3 + 2) = 0

## Conclusion

In this article, we have discussed the definition of limit calculus with its example. We described the important

properties of limit. We have covered different types of limits in this article. We learned various methods to

evaluate the limit problems. We did solve some examples for you to understand the limit better manner. After

reading this article, you can calculate the limit easily.