# Introduction to Limits with its definition, types, examples, and rules

Limit is an essential topic in Calculus, it is used in almost all important and main topics like differentiation and integrals. Limits refer to something that confines, restricts, and bounds.

In mathematics, the concept of limit is based on closeness, it is normally used to assign values to some functions at that point at which no values are defined. In calculus, a limit is a value on which a function approaches the output.

The history of this mathematical concept is very old and authentic. Limits were firstly introduced by the Italian mathematician Archimedes almost 2300 years ago. He introduced the idea of limits to measure curved surfaces and figures.

In this post, we’ll study the basic definition, rules & types of limits with examples.

## Definition:

According to Wikipedia:

“In mathematics, a limit is a value that a function (or sequence) approaches as the input (or index) approaches some value”

## Formula of limit:

The formula of limit is usually written as:

We can read it as the limit of f(x) as x approaches c equal to L

• f(x) is the function.
• c is the limit value.

## Types of limits:

Limit has three types.

• Right-hand limit
• Left-hand limit
• Two-sided limit

### Right-hand limit:

The right-hand limit is an approaching value of f(x) as the x approaches c from the right side.

The plus sign represents that the limit value comes from the right and must be greater than c.

### Left-hand limit:

The approaching value of a function F(x) as the x approaches c from the left side is called the left-hand limit.

The negative symbol represents that the limit term comes from the left and must be less than c.

### Two-sided limit:

The two-sided limit represents that the left-hand limit and right-hand limit exist and both have the same result. No sign is used for the two-sided limit.

## How to calculate limits in calculus?

To evaluate the limit, follow the below examples.

Example 1: Calculate the limit of 12x3 – 5x2 + 21x + 32 / x5, when x approaches 2.

Solution:

Step 1: Take the given function and apply the limits.

f(x) = 12x3 – 5x2 + 21x + 32 / x5

c = 2

Step 2: Apply the limits notation separately by using the sum, difference, and quotient rules

Step 3: Write the constants outside the limits.

Step 4: Now put the limit value.

To get rid of these long calculations you can use an online limit calculator with steps.

### How to use the limit calculator:

1. Enter the function in the input box.
2. Select the variable.
3. Select the limit type.
4. Enter the limit.
5. Cross-check your input box from the display box.
6. Hit calculate button.

Example 2:

If the function is not giving a finite answer after putting the limits then we use the L’hospital rule to solve that kind of function.

Example: Let us consider a function f(x) = (3x3 – 9x2) / 3x – 9 as x approaches to 3.

Solution:

Step 1: Take the given function and apply the limits.

F(x) = (3x3 – 9x2) / 3x – 9

c = 3

Step 2: Now apply the value of the limit.

Step 3: To remove the indeterminate form apply the L’hospital rule. Take the derivative of the upper and lower side of the function separately.

d/dx F(x) = d/dx (3x3 – 9x2) / d/dx (3x – 9)

F(x) = (9x3-1 – 18x2-1) / (3(1) – 0)

F(x) = (9x2 – 18x) / 3

F(x) = 3x2 – 6x

Step 4: Now apply the limit value again.

## Summary:

In the above article, we have studied the basic definition of Limit, its types, and the basic history of limits. After reading this article carefully you will be able to solve almost all the problems related to this topic. The basic intent of the L’hospital rule is discussed above.

## Mean Value Theorem for Higher Dimensions

Let $f$ be differentiable on a connected set $E\subseteq \mathbb{R}^n$, then for any $x,y\in E$, there exists $z\in E$ such that $f(x)-f(y)=\nabla f(z)\cdot (x-y)$.

Proof: The trick is to use the Mean Value Theorem for 1 dimension via the following construction:

Define $g:[0,1]\to\mathbb{R}$, $g(t)=f(tx+(1-t)y)$. By the Mean Value Theorem for one variable, there exists $c\in (0,1)$ such that $g'(c)=\frac{g(1)-g(0)}{1-0}$, i.e.

$\nabla f(cx+(1-c)y)\cdot (x-y)=f(x)-f(y)$. Here we are using the chain rule for multivariable calculus to get: $g'(c)=\nabla f(cx+(1-c)y)\cdot (x-y)$.

Let $z=cx+(1-c)y$, then $\nabla f(z)\cdot (x-y)=f(x)-f(y)$ as required.

## The Scientific (Mathematical) Way to Cut a Cake

Ever wondered if there is an alternative way to cutting cake so that it can stay fresh and softer in the refrigerator?

This is how!

For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it.

All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner’s popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A’s but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an “inner monologue”–the train of thought students should be following in order to solve the problem–providing the necessary reasoning as well as the solution. The book’s emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory.

The Calculus Lifesaver combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus.

• Serves as a companion to any single-variable calculus textbook
• Informal, entertaining, and not intimidating
• Informative videos that follow the book–a full forty-eight hours of Banner’s Princeton calculus-review course–is available at Adrian Banner lectures
• More than 475 examples (ranging from easy to hard) provide step-by-step reasoning
• Theorems and methods justified and connections made to actual practice
• Difficult topics such as improper integrals and infinite series covered in detail
• Tried and tested by students taking freshman calculus

# Proving Quotient Rule using Product Rule

This is how we can prove Quotient Rule using the Product Rule.

First, we need the Product Rule for differentiation: $\displaystyle\boxed{\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}}$

Now, we can write $\displaystyle\frac{d}{dx}(\frac{u}{v})=\frac{d}{dx}(uv^{-1})$

Using Product Rule, $\displaystyle \frac{d}{dx}(uv^{-1})=u(-v^{-2}\cdot\frac{dv}{dx})+v^{-1}\cdot(\frac{du}{dx})$

Simplifying the above will give the Quotient Rule! :

$\displaystyle\boxed{\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}}$

You can also try proving Product Rule using Quotient Rule!