Understanding Limits in Calculus: Definition and Basic Rules

In mathematical calculus, a limit is a fundamental concept used to describe the behavior of a function as the input variable approaches a particular value. It is used to analyze the behavior of functions, such as their continuity and convergence. It plays a crucial role in the development of derivative and integral calculus.

It tells us the limit of a function can exist or not exist, be finite or infinite, and have a variety of different behaviors of the different functions. Nowadays, limits are not used only in calculus but also in many other branches of mathematics and science, including differential equations, physics, and engineering. 

In this article, we will discuss the detailed definition of a limit and the basic formulas of a limit. For a better understanding of limit solving different examples.

Limit’s Definition: Exploring the Fundamental Concept

The limit of a function g(X) as “X” approaches a value “c” is denoted by. 

Lim _X→c g(X) = L  

Where “L” is the limiting value of the function.

This notation indicates that we are interested in the behavior of g(X) as “X” gets closer and closer to “c”, but not necessarily equal to c. The limit can exist even if the function is not defined at c or if it is defined differently at c.

Limits are played a crucial role in calculus and mathematical analysis, and they are used to define important concepts such as continuity, derivatives, and integrals. It also has practical applications in fields such as physics and engineering.

Extended Definition of Limit: Epsilon-Delta Approach

The extended definition of a limit involves the concept of neighborhoods and epsilon-delta definitions. The extended definition of a limit is defined as:

Let g(X) be a function defined on an interval that is open and contains the point “c,” perhaps with the exception of “c” itself. If for every ε > 0, there exists a δ > 0 such that,

 If |X – c| < δ, then |g(X) – L| < ε.

In this definition, ε represents a positive number that specifies how the function values are nearer to the limit L. “δ” represents a positive number that suggests how close the input values “X” to “c” are in order for the function values with “ε” of the limit “L”.

This definition is important because it provides a way to describe the behavior of functions as their input values approach a certain point. It is used to prove many theorems in calculus and is performed many developments in modern analysis of mathematics and engineering.

Basic Rules of Limit: A Comprehensive Guide

In this section, we discussed the basic rules of limits.

• Constant Rule: The constant function’s limiting value is identical to that of the constant itself. If “b” is a constant, then

 Lim _X→c (b) = b.

• Identity Rule: The limit of the identity function (f(X) = X) as “X” approaches “c” then its limiting value is “c”. 

  Lim _X→c (x) = c.

• Sum Rule: The limit of the sum of two functions is equal to the sum of their limits. That is, if  Lim _X→c (f(X)) = L and Lim _X→c (g(X)) = M, then

        Lim _X→c [f(X) + g(X)] = L + M.

• Difference Rule: The limit of the difference between two functions is equal to the difference of their limits. That is, if  Lim _X→c (f(X)) = L and Lim _X→c (g(X)) = M, then

Lim _X→c [f(X) –  g(X)] = L – M.

• Product Rule: The limit of the product of two functions is equal to the product of their limits. If Lim _X→c (f(X)) = L and Lim _X→c (g(X)) = M, then

              Lim _X→c [f(x)g(x)] = LM.

• Quotient Rule: The limit of the quotient of two functions is equal to the quotient of their limits (the denominator is not zero). That is, if Lim _X→c (f(X)) = L and Lim _X→c (g(X)) = M (where M ≠ 0), then 

Lim _X→c [f(x)/g(x)] = L/M.

• Power Rule: The limit of a power function (f(X) = Xⁿ, where n is a positive integer) as “X” approaches “c” is equal to cⁿ. That is,  

Lim _X→c [f(X)] = Lim _X→c (Xⁿ) = cⁿ.

• Exponential Rule: The limit of the exponential function (f(X) = e^X) as “X” approaches “c” then the limiting value is “e^c”. That is, 

Lim _X→c [f(X)] = Lim _X→c (e^X) = e^c.

Example of Limits: Solving Real-World Problems

In this section, we’ll discuss the different examples to understand how to find the limit of a function.

• Example 1:

Find the solution of the given function by the limits formula x² – 5 at x = 3.

Solution:

1. Step 1: let the given value is equal to G(X).

G(X) = X² – 5

2. Step 2: Apply the limit on both sides and put the limiting value carefully.

 Lim _X→3 G(X) = Lim _X→3 (X² – 5)

3. Step 3: With the help of the difference rule of limit we get.

     Lim _X→c [f(X) – g(X)] = Lim _X→c [f(X)] – Lim _X→c [g(X)]

Lim _X→3 G(X) = Lim _X→3 (X²) – Lim _X→3 (5)

4. Step 4: Apply the constant, and identity rule and also simplify.

     Lim _X→c X = c, Lim _X→c b = b  

Lim _X→3 G(X) = [(3)² – (5)] = 9 – 5 = 4

Lim _X→3 G(X) = 4 is the solution of X² – 5 at x = 3.

• Example 2:

Find the solution of the given function by the limits formulas, X² – 3X + 4 at x = -2.

Solution:

1. Step 1: let the given value is equal to G(X).

G(X) = X² – 3X + 4

2. Step 2: Apply the limit on both sides and put the limiting value carefully.

 Lim _X→-2 G(X) = Lim _X→-2 (X² – 3X + 4)

3. Step 3: With the help of the difference rule of limit we get.

     Lim _X→c [f(X) – g(X) + h(X)] = Lim _X→c [f(X)] – Lim _X→c [g(X)] + Lim _X→c [h(X)]

Lim _X→-2 G(X) = Lim _X→-2 (X²) – Lim _X→-2 (3X) + Lim _X→-2 (4)

4. Step 4: Apply the constant and identity rule and also simplify.

     Lim _X→c X = c, Lim _X→c b = b  

Lim _X→-2 G(X) = [(-2)² – 3(-2) + (4)] = 4 + 6 + 4 = 14

Lim _X→-2 G(X) = 14 is the solution of X² – 3X + 4 at x = -2.

Summary:

In this article, we discussed the definition of a limit and the basic rules of the limit such as product, sum, difference, and quotient. To understand the idea of limit solved the examples with the help of constant, difference, and sum rules in detail.