Limits Formula

What is Limit ?

Limits examples are one of the most difficult concepts in Mathematics according to many students. However, through easier understanding and continued practice, students can become thorough with the concepts of what is limits in maths, the limit of a function example, limits definition and properties of limits. Limits math is one of the most important concepts in Calculus. Calculus is a branch of mathematics that deals with the calculations related to continuously changing quantities. Math limit formula can be defined as the value that a function returns as an output for the given input values.


What are Limits & Limits Formula in Maths?

Limits math is very important in calculus. It is one of the basic prerequisites to understand other concepts in Calculus such as continuity, differentiation, integration limit formula, etc. Most of the time, math limit formula are the representation of the behaviour of the function at a specific point. Hence, the concept of limits is used to analyze the function. The mathematical concept of Limit of a topological net generalizes the limit of a sequence and hence relates limits math to the theory category. Integrals in general are classified into definite and indefinite integrals. The upper and lower limits are specified in the case of definite integration limit formula. However, indefinite integration limit formula are defined without specified limits and hence have an arbitrary constant after integration. The subsequent sections elaborate a brief overview of various concepts involved in a better understanding of math limit formula.

Limits formula:- Let y = f(x) as a function of x.  If at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. If these values tend to some definite unique number as x tends to a, then that obtained a unique number is called the limit of f(x) at x = a. 


Limits Math

The limit of a real-valued function ‘f’ with respect to the variable ‘x’ can be defined as:

\[\lim_{x \to p}f(x)\] = L

In the above equation, the word ‘lim’ refers to the limit.

It generally describes that the real-valued function f(x) tends to attain the limit ‘L’ as ‘x’ tends to ‘p’ and is denoted by a right arrow. 

We can read this as: “the limit of any given function ‘f’ of ‘x’ as ‘x’ approaches to ‘p’  is equal to ‘L’”.


What are the Properties or Laws of Limits?

The Properties of Limits Are as Follows:

  1. The Notation of a Limit

The limit of a function is denoted as f (x) → L as x → p or in the limit notation as:

\[\lim_{x \to p}f(x)\] = L

Let us assume that there exists \[\lim_{x \to p}f(x)\], \[\lim_{x \to p}g(x)\], \[\lim_{x \to p}f_{1}(x)\], ….. \[\lim_{x \to p}f_{n}(x)\]. This assumption is made to explain the other properties of limits.


  1. The Sum Rule

The sum rule states that the sum of the individual limits of any two functions is equal to the limit of the sum of those functions. 

\[\lim_{x \to p}f(x)\] + \[\lim_{x \to p}g(x)\] = \[\lim_{x \to p}\] | f(x) + g(x) |


  1. The Extended Sum Rule

The extended sum rule is the same as the sum rule. However, it is defined for the limits of more than two functions. 

\[\lim_{x \to p}\] f1(x) + \[\lim_{x \to p}\] f2(x) + …..\[\lim_{x \to p}\] fn (x) = \[\lim_{x \to p}\] | f1(x) + f2(x) + ……+ fn(x) |


  1. The Constant Function Rule

The constant function rule states that the limit of a constant function is equal to a constant.


  1. The Constant Multiple Rule

The limit of a function, when multiplied by a constant value, is equal to the constant multiplied by the limit of the function. 

\[\lim_{x \to p}\] k f(x) = k \[\lim_{x \to p}\] f(x)


  1. The Product Rule

The product rule states that the product of the limits of two individual functions is equal to the limit of the product of the functions.

\[\lim_{x \to p}\] f(x) * \[\lim_{x \to p}\] g(x) = \[\lim_{x \to p}\] |f(x) * g(x)|


  1. The Extended Product Rule

The extended product rule is the same as the product rule. However, more than two functions are taken into consideration.

\[\lim_{x \to p}\] f1(x) * \[\lim_{x \to p}\] f2(x) * …..* \[\lim_{x \to p}\] fn(x) = \[\lim_{x \to p}\] |f1(x) * f2(x) * ….*fn(x)|


  1. The Quotient Rule

The quotient of individual limits of two functions when the limit of the denominator is not equal to zero is equal to the limit of the quotient of the two functions where the denominator function is not equal to zero.

\[\frac{\lim_{x \to p} f(x)}{\lim_{x \to p} g(x)}\] = \[\lim_{x \to p}\] \[\frac{f(x)}{g(x)}\]


  1. The Power Rule

The power rule of limits is Mathematically stated as follows:

\[\lim_{x \to p}\] |f(x)|k = [ \[\lim_{x \to p}\] f(x) ]k

where ‘k’ is known to be any an integer

Similarly, when the powers are fractions, the power rule can be stated as:

\[\lim_{x \to p}\]  \[\sqrt{|f(x)}|\] = \[\sqrt{\lim_{x \to p}f(x)}\]


Important Limit Formula List in Limits 

Math Limit Formula List

1

\[\lim_{x \to 0}sin x = 0\]

2

\[\lim_{x \to 0}cos x = 1\]

3

\[\lim_{x \to 0}sin \;x/x = 1\]

4

\[\lim_{x \to 0}log\;(1+x)/x = 1\]

5

\[\lim_{x \to 0}log\;e^{x} = 1\]

6

\[\lim_{x \to e}log\;e^{x} = 1\]

7

\[\lim_{x \to 0}log\;e^{x} - 1/x = 1\]

8

\[\lim_{x \to 0}log\;a^{x} - 1/x = log_{e^{a}}\]


Above is the limit formula list.


Limits Examples and Solutions

Question 1) Evaluate \[\lim_{x \to 2}\] (1x3 - 3x2 + 6x -3)

Solution:

\[\lim_{x \to 2}\] (1x3 - 3x2 + 6x -3)

= \[\lim_{x \to 2}\](1x3) -  \[\lim_{x \to 2}\] (3x2) +  \[\lim_{x \to 2}\] (6x) -  \[\lim_{x \to 2}\] (3)

= 1 \[\lim_{x \to 2}\](x3) -  3 \[\lim_{x \to 2}\] (x2) + 6 \[\lim_{x \to 2}\] (x) -  (3)

= 1(2)3 - 3(2)2 +6(2) -3

= 1 x 8 - 3 x 4 + 12 - 3 

= 8 - 12 + 9

= 17-12

= 5

FAQ (Frequently Asked Questions)

1. What is the Formula of Limits?

What is Limit? Let y = f(x) as a function of x. If at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. If these values tend to some definite unique number as x tends to a, then that obtained a unique number is called the limit of f(x) at x = a.

2. Who Invented Limits?

Archimedes' thesis, The Method, was lost until 1906, when mathematicians discovered that Archimedes came close to discovering infinitesimal calculus. As Archimedes' work was not known until the twentieth century, then others developed the modern mathematical concept of limits.

3. What is the Limit of a Constant?

The limit of constant times a function is equal to the constant times the limit of the function. We know that the limit of a product is always equal to the product of the limits. The limit of a quotient is equal to the quotient of the limits. The limit of any given constant function is equal to the constant.

4. What is Infinity Minus Infinity?

It is impossible for infinity subtracted from infinity to be equal to one and zero. Using such type of math, we can get infinity minus infinity as equal to any real number. Therefore, infinity subtracted from infinity is undefined.