Limit Calculus- Explained with Its Definition Types and Calculations

In mathematics, a limit is a wide concept that is used to calculate the numerical value of the function at a specific point. It is frequently used in mathematical analysis for defining differential, integral, Taylor series, and continuity.

There are several types of limits: one-sided limits, two-sided limits, infinite limits, and limits at infinity. In this blog post, we'll discuss all of these types of limits and how to calculate them. We'll also touch on the squeeze theorem and its applications. By the end of this post, you should have a good understanding of limits and how to calculate them.

What is the limit in calculus?

A limit is simply a boundary within which something may or may not exist. It is important to understand what constitutes a limit before attempting to solve a problem involving it.

The limit of a function f(u) as “u” approaches “b” is equal to N and is said to be the general expression of the limit calculus. Mathematically it can be written as:

Lim_u_→b f(u) = N

u = the independent variable of the function
b = the particular point of the function.
f(u) = the given function
N = the numerical result of the function

Types of limit calculus

There are two main types of limits:

One-sided (left hand or right hand)
Two-sided limits

Let us understand these types of limits briefly.

One-sided limit

With one-sided limits, once an input value reaches or exceeds the limit, everything after that point is also considered to be beyond that limit (the function "folds over"). For example, consider a function that determines how many integers are larger than a given number.

x > y means x ≤ y+1 and every integer between x and y inclusive is larger than y + 1. Thus x = 5, 6, 7, 8 would all be true but 9 wouldn't because 9 > 5 + 1.

Two-sided limit

With two-sided limits, however, no matter how close an input value gets to the limit, there's still a chance that future inputs might exceed it (the function "jumps over"). For example, consider our previous function which determines whether or not an integer is larger than a given number.

x ≥y means either x ≤y+1 or else x ∈ [y+1.x]. As long as no integer falls within [y+1.x], then x ≥ y. So 10 would always be false but 11 would be true since 11 ∈ [10. 12].

How to calculate the limit calculus?

There are various ways to calculate the problems of limit calculus such as by rationalization, factorization, and using its laws. Let us take a few examples of limit calculus.

Example 1

Calculate the limit of the given function if the specific point is 4.

f(u) = 3u² + 5u – 4u³ / u² * 12u – 6

Solution

Step 1: First of all, take the given function and use the notation of limit to write the function according to the general expression of the limit.

f(u) = 3u² + 5u – 4u³ / u² * 12u – 6

lim_u_→_b [f(u)] = lim_u_→₄ [3u² + 5u – 4u³ / u² * 12u – 6]

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

lim_u_→₄ [3u² + 5u – 4u³ / u² * 12u – 6] = lim_u_→₄ [3u²] + lim_u_→₄ [5u] – lim_u_→₄ [4u³] / lim_u_→₄ [u²] * lim_u_→₄ [12u] – lim_u_→₄ [6]

Step 3: Apply the constant function rule of limit calculus.

lim_u_→₄ [3u² + 5u – 4u³ / u² * 12u – 6] = 3lim_u_→₄ [u²] + 5lim_u_→₄ [u] – 4lim_u_→₄ [u³] / lim_u_→₄ [u²] * 12lim_u_→₄ [u] – lim_u_→₄ [6]

Step 4: Now to evaluate the limit result, substitute 4 in the place of “u” in the above expression.

lim_u_→₄ [3u² + 5u – 4u³ / u² * 12u – 6] = 3 [4²] + 5 [4] – 4 [4³] / [4²] * 12 [4] – [6]

lim_u_→₄ [3u² + 5u – 4u³ / u² * 12u – 6] = 3 [4x 4] + 5 [4] – 4 [4 x 4 x 4] / [4 x 4] * 12 [4] – [6]

lim_u_→₄ [3u² + 5u – 4u³ / u² * 12u – 6] = 3 [16] + 5 [4] – 4 [64] / [16] * 12 [4] – [6]

= 48 + 20 – 256 / 16 * 48 – 6

= 48 + 20 – 16 * 48 – 6

= 48 + 20 – 768 – 6

= 68 – 768 – 6

= -700 – 6 = -706

You can get help through a limit calculator to avoid time-consuming calculations.

Example 2

Calculate the limit of the given function if the specific point is infinity.

f(z) = (2z² – 5z + 3)/(z² – 9)

Solution

Step 1: First of all, take the given function and use the notation of limit to write the function according to the general expression of the limit.

f(z) = (2z² – 5z + 3)/(z² – 9)

lim_z_→_b [f(u)] = lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)]

Step 2: Now apply the notation of limit calculus separately to each function with the help of rules of limit calculus and evaluate the limit by substituting z equal to infinity.

Lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = (lim_z_→∞ [2z²] – lim_z_→∞ [5z] + lim_z_→∞ [3])/( lim_z_→∞ [z²] – lim_z_→∞ [9])]

lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = ([2(∞)²] – [5(∞)] + [3])/([∞²] – 9)

lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = (∞² – ∞ + 3)/(∞² – 9)

lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = ∞/∞

Step 3: Now try to factorize the function to make the function simpler.

Lim_z_→_b [f(u)] = lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)]

Lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = lim_z_→∞ [(2z² – 2z – 3z + 3)/(z² – (1)²)]

Lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = lim_z_→∞ [(2z(z – 1) – 3(z – 1)/((z – 1) (z + 1))]

Lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = lim_z_→∞ [(2z – 3)(z – 1)/((z – 1) (z + 1))]

Lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = lim_z_→∞ [(2z – 3)~~(z – 1)~~/(~~(z – 1)~~ (z + 1))]

Lim_z_→∞ [(2z² – 5z + 3)/(z² – 9)] = lim_z_→∞ [(2z – 3)/(z + 1)]

Step 4: Now take “z” common from the above expression and again apply the limit value.

= lim_z_→∞ [z(2 – 3/z)/z(1 + 1/z)]

= (2 – 3/∞)/(1 + 1/∞)

= (2 – 0)/(1 + 0) because something over infinity always give zero

= (2)/(1)

= 2

Example 3:

Calculate the limit of the given function if the specific point is 2.

f(w) = (w³ – 7w² + 20) / (2w² – 8)

Solution

Step 1: First of all, take the given function and use the notation of limit to write the function according to the general expression of the limit.

f(w) = (w³ – 7w² + 20) / (2w² – 8)

lim_w_→_b [f(u)] = lim_w_→2 [(w³ – 7w² + 20) / (2w² – 8)]

Step 2: Now apply the notation of limit calculus separately to each function with the help of rules of limit calculus and evaluate the limit by substituting z equal to 2.

= (lim_w_→2 [w³] – lim_w_→2 [7w²] + lim_w_→2 [20]) / (lim_w_→2 [2w²] – lim_w_→2 [8])

= (lim_w_→2 [w³] – 7lim_w_→2 [w²] + lim_w_→2 [20]) / (2lim_w_→2 [w²] – lim_w_→2 [8])

= ([2³] – 7 [2²] + [20]) / (2 [2²] – [8])

= ([8] – 7 [4] + [20]) / (2 [4] – [8])

= (8 – 28 + 20) / (8 – 8)

= (-20 + 20) / (8 – 8)

= 0/0

Step 3: Now apply L’hopital’s rule of limit calculus as the function make an indeterminate form such as 0/0.

lim_w_→2 [(w³ – 7w² + 20) / (2w² – 8)] = lim_w_→2 [d/dw (w³ – 7w² + 20) / d/dw (2w² – 8)]

= lim_w_→2 [(3w² – 14w + 0) / (4w – 0)]

= lim_w_→2 [(3w² – 14w) / (4w)]

Apply the limit value again.

lim_w_→2 [(w³ – 7w² + 20) / (2w² – 8)] = lim_w_→2 [3w²] – lim_w_→2 [14w]) / (lim_w_→2 [4w])

= [3(2)²] – [14(2)]) / ([4(2)])

= [3(4)] – [14(2)]) / ([4(2)])

= [12 – 28]) / (8)

= -16 / 8

= -2

Bottom line

As we have seen, limits are a fundamental concept in mathematics that is used to define continuity, derivatives, and integrals. There are several types of limits: one-sided limits, two-sided limits, infinite limits, and limits at infinity. In this blog post, we have discussed all of these types of limits and how to calculate them along with examples.