In calculus, a limit is defined as a value that a function (or sequence) approaches as the input (or index)
approaches a particular value. It helps us to understand the behavior of functions and sequences near a
specific point or at infinity. Limits are considered the foundation of continuity, derivative, and series.
In this article, we will read the definition, types, and properties of limits. In addition, we will learn how to
calculate limits problems. Further, to gain a better understanding, we will solve different examples in the
example section.
Definition of Limit Calculus
Consider a function f(x) defined on an interval that contains the point c. We say that the limit of f(x) as x
approaches c is L, denoted by Lim x → c f(x) = L if, for any given positive value epsilon (ε > 0), there exists a
positive value delta (δ > 0) such that |f(x) – L| < ε for all x in the interval (x – δ, x + δ) that contain c.
Example:
Consider a function f(x) = 3x – 2. Find out the limit of the given function as x approaches 2.
Solution:
By using the definition of limit, we want to find L such that | (3x – 2) – L| < ε for all x satisfying 0 < | x – 2| < δ.
Substitute x = 2 into f(x), we get f(2) = 3(2) – 2 = 4
To ensure | (3x – 2) – 4| < ε, we choose δ = ε / 3.
Hence, the limit of f(x) as x approaches 2 is 4, and for any positive ε, we can find a corresponding δ = ε / 3 such
that | (3x – 2) – 4| < ε for all x satisfying 0 < | x – 2| < δ.
Properties of Limits
While dealing with limits, the following properties can help in simplifying calculations.
If Lim x→ a f(x), and Lim x→ a g(x) exist, and k is any constant then,
1. Lim x→ a [k f(x)] = k lim x→ a [f(x)]
2. Lim x→ a [f(x) ± g(x)] = Lim x→ a f(x) ± Lim x→ a g(x)
3. Lim x→ a [f(x). g(x)] = Lim x→ a f(x)). Lim x→ a g(x)
4. Lim x→ a (f(x) / g(x)) = Lim x→ a f(x) / Lim x→ a g(x) (provided lim x→ a g(x) ≠ 0)
5. Lim x→ a k = k
6. Lim x→ a (f(x)) k = (Lim x→ a f(x)) k
Types of Limit
There are several types of limits, but some common ones are given below:
- One-sided limits are limits of a function as its input approaches a specific point from one side only
(either left or right side). The left-hand limit is expressed as lim x → c – f (x), and the right-side limit is
written as lim x → c + f (x). - Two-sided limits are the most common type of limit. They describe the behavior of function from both
the left and right sides. - Limit at Infinity refers to the limits of a function when its input approaches positive or negative infinity,
expressed as lim x →± ∞ f(x) - An infinite limit occurs when the limit of a function tends to positive or negative infinity as the input
approaches a certain point, denoted as lim x → c f(x) = ± ∞. - Discontinuous limits happen when the limit of a function does not exist at a specific point due to a
break in the graph of the function.
Methods for Calculating the Limit
Here are some methods and techniques for calculating the limit of the function
1. Direct Substitution Method
When the function is defined at the value where the limit is being determined, then put this value into
the given function and calculate the result.
2. Factoring and canceling
When the function has an algebraic expression, you can try to find the factor and cancel the common
factors to simplify the given expression before calculating the limit.
3. Rationalization
When the function involves radical or complex fractions, you can use the rationalization method to
eliminate radicals and simplify the fraction.
4. Trigonometric Identities
You can use the following trigonometric identities if the function contains trigonometric functions.
- lim x→ 0 (1 – cos x / x) = 1
- lim x→ 0 (Sin x / x) = 1
- lim x→ 0 (tan x / x) = 1
5. Exponential and logarithmic limits
You can use the following formulas if the function contains exponential functions.
- lim x→ 0 (1 + x) (1/x) = e
- lim x→ ∞ (e x ) = ∞
- lim x→ – ∞ (e x ) = 0
6. L Hopital’s Rules
It evaluates limits of indeterminate forms (like 0/0, or ∞/∞). It involves taking the derivative of the
numerator and denominator individually and then evaluating the limit again.
Solved Examples of Limit
Here are some examples of finding limits with the help of limits rules manually.
Example 1.
Evaluate lim x→ 2 [(x 3 + 5x 2 + x + 5) / (x + 5)]
Solution
Lim x→ 2 [(x 3 + 5x 2 + x + 5) / (x + 5)]
= Lim x→ 2 [(x 2 (x + 5) + 1(x + 5)) / (x + 5)]
= Lim x→ 2 [(x 2 + 1) (x + 5)) / (x + 5)]
After canceling the common factor, we have
= Lim x→ 2 (x 2 + 1)
After applying the limit, we get
= 2 2 + 1 = 4 + 1 = 5
Hence, lim x→ 2 [(x 3 + 5x 2 + x + 5) / (x + 5)] = 5
Example 2.
Determine the lim x → ∞ (x 2 + 3) / (x 3 + 2)
Solution:
Lim x → ∞ (x 2 + 3) / (x 3 + 2) = ∞ / ∞
This is an indefinite form. Therefore, we will solve this problem with L Hopital’s Rules.
After taking the derivative of the numerator and denominator individually. We get
= Lim x → ∞ (2x / 3x 2 )
After cancellation, we have
= Lim x → ∞ (2 / 3x)
Now apply the limit
= 2 / 3(∞) = 2 / ∞ = 0 (∴1/ ∞ = 0)
Hence, the Lim x → ∞ (x 2 + 3) / (x 3 + 2) = 0
Conclusion
In this article, we have discussed the definition of limit calculus with its example. We described the important
properties of limit. We have covered different types of limits in this article. We learned various methods to
evaluate the limit problems. We did solve some examples for you to understand the limit better manner. After
reading this article, you can calculate the limit easily.