A Deep Dive into Limits: Exploring Properties and Examples in Calculus

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Calculus is a field of mathematics concerned with the study of change and motion. Limit is considered as a foundation of calculus. It describes the behavior of a function as the input values approach a particular value. Limit plays a crucial role in calculus for defining integrals, derivatives, and continuity.

In this article, we will explore the definition of the limit with its mathematical representation. We will discuss the different types of limits, properties, and methods to evaluate them. We will solve some examples to make it easier to understand.

Definition of Limit

Let a function f(x) be defined in an open interval near the number c (need not be at c). If, as x approaches c from both the left and right side of c, f(x) approaches a specific number L, then L is called the limit of f(x) as x approaches c. symbolically, it is written as:

limx → c f(x) = L

This notation can be read as a limit of function “f of x” as “x” approaches c is L. Here, the symbol lim represents limit.

Types of Limit

There are different types of limits based on how the input values approach a specific point. Limits can be classified into the following categories:

One-Side Limits:

One-sided limits are only approached from one side of the limit point. They are used to describe the behavior of a function as it approaches a limit point from the left or the right side.

A one-sided limit has two types:

  • Left-hand Limit
  • Right-hand limit

Left-Hand Limit

It represents how the function behaves as the input approaches c from values less than c. It can be written as:

limx → c – f (x)

Right-Hand Limit

The right-hand Limit or Limit from the Positive Side is denoted as limx → c+ f (x).

It signifies how the function behaves as the input approaches c from values greater than c.

Two-sided limit:

A two-sided limit describes the behavior of a function as it approaches a limit point from both the left and right sides. The two-sided limit can be expressed as:

limx → c f(x)

Infinite Limits:

A positive infinite limit indicates that the function grows without bound as x gets closer to c.

limx → c f(x) = ± ∞

Limit at Infinity:

It explains the behavior of the function as x becomes extremely small or extremely large.

limx →± ∞ f(x)

Properties of Limit

Properties of limit help us in solving complex problems by breaking them into simpler components. Let’s explore and learn more about these properties.

Limit of a Constant:

If k is a constant, then

limx→ c k = k

The limit of any constant value is equal to the constant itself.

Limit of a Sum or Difference:

The limit of the sum or difference of two functions is equal to the sum or difference of their respective limits.

lim x→ c [f(x) ± g(x)] = limx→ c f(x) ± limx→ c g(x)

Limit of a Product

The limit of a product is the product of the limits of the functions being multiplied.

limx→ c [f(x) × g(x)] = limx→ c f(x) × limx→ c g(x)

Limit of a Quotient

The limit of a quotient is the quotient of the limits of the numerator and denominator.

limx→ c [f(x) / g(x)] = (Limx→ c f(x)) / (Limx→ c g(x))

where limx→ c g(x) ≠ 0

Limit of a Power

The limit of a function raised to a constant power is the function raised to that power.

Limx→ c [f(x)]n = [Limx→ c f(x)]n

Here n is any arbitrary constant.

Techniques to Evaluate Limit

Several techniques can be used to find the limit of a function as it approaches a specific value or as it goes to infinity. Here are some common methods to evaluate limits:

Direct Substitution:

The simplest technique for evaluating limits is direct substitution. This technique can be used if the function is continuous at the point where the limit is being taken.

Example: To find the limit as x approaches 2 for the function f(x) = x3, substitute x = 2 into the function expression.

limx → 2 (x3) = 23 = 8.

Factoring and Canceling:

Sometimes, an expression can be simplified by factoring and canceling common factors. This is especially useful for rational functions.

Example: To find the limit as x approaches 3 for the function f(x) = (x2 – 9) / (x – 3), factor the numerator as (x + 3) (x – 3) and cancel out the common factor of (x – 3) to simplify the expression to limx → 3 (x + 3).

L’Hôpital’s Rule:

It states that if the limit of the form limx → c [f(x) / g(x)] is indeterminate (0/0 or ∞/∞) then take the derivative of the numerator and denominator separately and then evaluate the limit again. This rule can be applied repeatedly if necessary.

You can try online calculators like a limit solver by Allmath to evaluate limits according to the above-mentioned techniques with steps.

Special Limits:

Here are some examples of special limits that have known values without needing to apply advanced techniques:

Examples of Limits

Here are some examples of limits along with their solutions:

Example 1:

Find the limit as x approaches 2 for the function f(x) = 3x – 1.


Directly substitute the value into the expression:

limx→ 2 (3x – 1) = 3(2) – 1 = 6 – 1 = 5.

Thus, limx→ 2 (3x – 1) = 5.

Example 2:

Evaluate limx→ 1 (x2 – 1) / (x – 1).


limx→ 1 (x2 – 1) / (x – 1) = 0/0 (indeterminate form)

(x2 – 1) can be factored as (x + 1)(x – 1) using the difference of squares formula.

Now, rewrite the limit with the factored numerator:

limx→ 1 (x2 – 1) / (x – 1) = limx→ 1 (x – 1)(x + 1) /(x – 1)

Notice that there is a common factor of (x – 1) in both the numerator and the denominator. Cancel out this common factor:

limx→ 1 (x2 – 1) / (x – 1) = limx→ 1 (x + 1)

Now, substitute x = 1:

limx→ 1 (x + 1) = 2

Hence, limx→ 1 (x2 – 1) / (x – 1) = 2.

Example 3:

Evaluate limx→ 0 (ex – 1)/x


limx→ 0 (ex – 1)/x = 0/0

Apply L’Hôpital’s Rule:

Take the derivatives of the numerator and denominator separately.

limx→ 0 (ex )/1 = 1

Thus, limx→ 0 (ex – 1)/x = 1


In this article, we have discussed the definition of a limit, its various types of limits, and delved into their properties. We have covered essential techniques to evaluate limits effectively. The provided examples have illustrated these concepts in practice, enhancing your understanding of this crucial mathematical tool.

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I am a mechanical engineer by profession. Just because of my love for fundamental physics, I switched my career, and therefore I did my postgraduate degree in physics. Right now I am a loner (as ever) and a Physics blogger too. My sole future goal is to do a Ph.D. in theoretical physics, especially in the field of cosmology. Because in my view, every aspect of physics comes within the range of cosmology. And I love traveling, especially the Sole one.

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