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

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.
  • L is the answer
  • 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.

Some basic rules of 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!


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Proving Quotient Rule using Product Rule

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!