In mathematics, limit calculus is a fundamental concept that allows studying the behavior of functions as the given function approaches a specific point. The concept of limit calculus has wide use in finance, physics, engineering, and economics.

The limit is used in other branches of calculus to explain and calculate differential, integral, and continuity of functions. In this article, we’ll explain the definition, rules, and solved examples of limit calculus.

What is Limit Calculus?

In calculus, the limit of a function is a value that a function (f(x)) approaches as the independent variable “x” approaches a particular point. It is denoted by “lim, Lim, or Lt”, and shows the behavior of the function as it gets closer and closer to a specific point.

The mathematical expression of limit calculus is:

Lim­x→b f(x) = L

The above expression can be read as the function f(x) has a limit “L” at x = b. The epsilon definition of limit calculus is:

The limit of a function will exist at a particular point iff for each positive number ɛ, there must hold a positive number δ, such that

If 0 < |x-a| < δ

Then, |f(x)-L| < ɛ.

Rules of Limit Calculus

Here are some specific rules of limit calculus.

Applications of Limit Calculus

Limit calculus has vast applications in mathematics, engineering, and science. Some of the commonly used applications of limit calculus are:

How to calculate limit problems?

Here is an example to learn how to evaluate the limit problems.

Example

Find the limit of 5x2 + 3x – 12 * 3x3 / 3x2 when x approaches 3.

Solution

Step 1: Take the given algebraic expression and write it according to the general expression of limit calculus.

Lim­x→3 [5x2 + 3x – 12 * 3x3 / 3x2]

Step 2: Now apply the sum, difference, product, and quotient rules of limit calculus to the above algebraic expression.

Lim­x→3 [5x2 + 3x – 12 * 3x3 / 3x2] = Lim­x→3 [5x2] + Lim­x→3 [3x] – Lim­x→3 [12] * Lim­x→3 [3x3] / Lim­x→3 [3x2]

Step 3: Now apply the constant multiple rules to the above expression.

Lim­x→3 [5x2 + 3x – 12 * 3x3 / 3x2] = 5Lim­x→3 [x2] + 3Lim­x→3 [x] – Lim­x→3 [12] * 3Lim­x→3 [x3] / 3Lim­x→3 [x2]

Step 4: Now apply the power rule and constant rules.

Lim­x→3 [5x2 + 3x – 12 * 3x3 / 3x2] = 5 [Lim­x→3 x]2 + 3 [Lim­x→3 x] – Lim­x→3 [12] * 3 [Lim­x→3 x]3 / 3 [Lim­x→3 x]2

Step 5: Apply Limit value x = 3

= 5 [3]2 + 3 [3] – [12] * 3 [3]3 / 3 [3]2

= 5 [9] + 3 [3] – [12] * 3 [27] / 3 [9]

= 45 + 9 – 12 * 81 / 27

= 45 + 9 – 972 / 27

= 45 + 9 – 36

= 18

Conclusion

The limit is the fundamental concept in calculus to deal with the behavior of the functions at a particular point.

The rules of the limit are very essential to evaluate the limit of the function. The applications of limit calculus are very vast.