Limit calculus is a fundamental branch of mathematics that is essential for integral calculus and differential calculus. With the help of limit calculus, mathematicians come to or are able to understand the better behavior of functions at a specific point.

The limit calculus is helpful for determining the behavior of a function at a certain point. It is a critical branch of calculus as it is helpful in defining continuity, derivative, and integral in various ways.

What is limit calculus?

Calculus of limits is a fundamental branch of calculus that deals with the study of limits at certain points and its basics properties. The definition of a limit calculus is “a point at which the function comes closer and closer to a certain point”.

The behavior of a function is described with the help of limit calculus as a certain point approaches a specific point or infinity. Limit is used in differential calculus as finding the derivative of a function with the first principle method.

The general equation of limit calculus with respect to independent variables can be written as :

Lim_v→s f(v) = M

* Lim is the notation of limit calculus
* v is the independent variable of the function
* s is the specific point of limit calculus
*  f(v) is the function
*  M is the numerical value of the function

Types of limit calculus :

There are three well-known types of limit calculus :

Here is a brief introduction to the basic types of limit calculus.

Left-hand limit :

Left-hand limit of a function f(v) as “v” approaches from left and is denoted by:

Lim_v→s^- f(v) = M

the function f(v) gets closer and closer to the specific point from the left side i.e., (for x<s).

Right-hand limit :

Right-hand limit of a function f(v) as “v” approaches from right and is denoted by:

Lim_v→s⁺ f(v) = M

The function f(v) gets closer and closer to the specific point from the right side i.e., (for x>s).

Two-sided limit :

Two hand limit of a function f(v) as “v” approaches from both sides and is denoted by:

Lim_v→s f(v) = M

The function f(v) gets closer and closer to the specific point from both sides i.e., (for x>s and x<s). In general, when both left and right-hand limits exist then the two-sided limit holds.

How to calculate the problems of limit calculus?

The problems of limit calculus can be evaluated with the help of laws and the general expression of limit calculus. Alternatively, a limit solver can be a handy tool to calculate the problems of limit calculus with steps.
Let us take an example of limit calculus to learn how to evaluate it manually.

Example :
Evaluate the limit of the given function at “2”
f(v) = 3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v

Solution :

Step 1 : Apply the notation of limit “lim” to the given function along with a specific point.

p(u) = 5u³ + 3u² – 8u * 12u⁴ – 3u² + 2
Lim_v→s [f(v)] = lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v]

Step 2 : Now apply the rules of limit calculus to write the notation of limit with each term of the function.

lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = lim_v→2 [3v²] – lim_v→2 [4v⁵] * lim_v→2 [2v] + lim_v→2 [9v⁴] + lim_v→2 [4v³] + lim_v→2 [2v]

Step 3 : Take out the constant coefficients of each term of the function.

lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = 3lim_v→2 [v²] – 4lim_v→2 [v⁵] * 2lim_v→2 [v] + 9lim_v→2 [v4⁴] + 4lim_v→2 [v³] + 2lim_v→2 [v]

Step 4 : Now put 2 to the independent variable of the function.

lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = 3 [2²] – 4 [2⁵] * 2 [2] + 9 [2⁴] + 4 [2³] + 2 [2]
lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = 3 [4] – 4 [32] * 2 [2] + 9 [16] + 4 [8] + 2 [2]
lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = 12 – 128 * 4 + 144 + 32 + 4
lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = – 116 * 4 + 144 + 32 + 4
lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = -464 + 144 + 32 + 4
lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = -320 + 32 + 4
lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = -288 + 4
lim_v→2 [3v² – 4v⁵ * 2v + 9v⁴ + 4v³ + 2v] = -284

Sum Up :

Limit calculus is one of the fundamental branches of mathematics that deals with the behavior of the function at a certain point. There are three well-known branches of limit calculus. The calculation of limit calculus can be done with the help of rules and general expression of limit.