A Limit in Mathematics: Definition, Types, and Examples

What is a limit in mathematics?

Suppose f(x) is a function that is defined in a specific interval around a point c, except possibly at c itself. The limit of f(x) as x approaches c is thus L, written lim x → c = L. For any positive number ε (ε > 0), there exists a positive number δ (δ > 0) such that |f(x) – L| < ε for all x in the interval (c – δ, c + δ) except possibly at c.

In other terms, for any small positive value epsilon (ε), we can find a small positive value delta (δ) such that whenever the distance between x and c is less than δ, the distance between f(x) and L is less than ε.

Types of Limit

There are many types of limits, but some of the most common types are:

1.     One-sided limit: A one-sided limit is a function’s limit when its input approaches a certain value from either the left or right side limit. It allows us to examine the behavior of the function specifically from one direction. It is denoted as lim x → c– or x → c+, representing the limit as x approaches c from the left side or right side, respectively.

2.     Two-sided limit: A two-sided limit is the limit of a function f(x), as it approaches a specific value c from both the left and right sides simultaneously. For a function f(x), the two-sided limit as x approaches a specific value c, denoted as lim x → c f(x), requires evaluating the behavior of the function as x approaches c from both from the left side (x < c), and from the right side (x > c).

3.     Infinite Limit: Infinite limit occurs when the value of a function approaches positive or negative infinity as the input approaches a certain value. It is written as lim x → c f(x) = ± ∞.

4.     Limit of Infinity: Limits at infinity describe the behavior of a function as the input values become arbitrarily large or small. It can be written as lim x → ∞ or lim x → – ∞, expressing the limit as x approaches positive or negative infinity, respectively.

5.     Discontinuities: Discontinuities are points at which a function is not continuous.

Characteristics of limit 

A few essential properties of the limit are given below:

Addition and Subtraction properties of limit

The sum or difference of the limits of two functions is equal to the sum or difference of their sum. Mathematically, if lim x→ c f(x) and lim x→ c g(x) exist then,

                             Lim x→ c [f(x) ± g(x)] = Lim x→ c [f(x)] ± Lim x→ c [g(x)]

Constant multiple properties

The limit of a function multiplied by a constant is equal to the constant multiplied by the limit of the function. Mathematically, if lim x→ c f(x) exists then, and k is a content then
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                                                Lim x→ c [k f(x)] = k lim x→ c [f(x)]

Product property of limit

Mathematically, if lim x→ c f(x) and lim x→ c g(x) exist then,

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

Quotient Property of limit

The limit of the quotient of two functions is equal to the quotient of their limit, provided the limit of the denominator is not zero. Mathematically, if lim x→ c f(x) and lim x→ c g(x) exist, and lim x→ c g(x) ≠ 0 then,

                                 Lim x→ c [f(x) / g(x)] = Lim x→ c [f(x)] / Lim x→ c [g(x)]

Exponent (power) property of limit

Mathematically, if lim x→ c f(x) exists then, and k is content, then

                                              Lim x→ c [f(x)] n = [Lim x→ c f(x)] n

These properties are useful for evaluating limits and performing calculations involving limits of function.

Theorems for limit:

There are a few extraordinary rules that can help evaluate limits.

·       Squeeze theorem (Sandwich Theorem): Let f, g, and h be functions such that f(x) ≤ g(x) ≤ h(x) for all x in an open interval containing c. If the limit of f(x) and h(x) exist and both are equal to a common value L, then the limit of g(x) also exists and is equal to L.

·       L’ Hopital’s Rule: It is used when we have an indeterminate form; such as 0 / 0 or ∞ / ∞ in limit. This rule provides a method to find the limit by taking the derivative of the numerator and denominator of the function.

·       Limit of Trigonometric functions: There are specific limits involving trigonometric functions that have well-known values. Like lim x→ 0 (Sin x / x) = 1, lim x→ 0 (tan x / x) = 1, and lim x→ 0 (1 – cos x / x) = 1.

·       Exponential and logarithmic limits: There are also specific limits involving exponential and logarithmic functions. For example, lim x→ 0 (1 + x) (1/x) = e.

You can take assistance from a limit calculator by Allmath to find the limit value of the given function according to the theorems of limit calculus.

Solved example of Limit

Example

Determine the limit of the function f(x) = [sin (4x) / x] as x approaches 0.

Solution

We will find lim x→ 0 (Sin 4x / x)

 Lim x→ 0 (Sin 4x / x) = lim x→ 0 (4Sin 4x / 4x) (By multiplying and dividing 4)

Apply the constant multiple rules of limit

= 4 lim x→ 0 (Sin 4x / 4x)

∴ Lim x→ 0 (Sin x / x) = 1

= 4(1) = 4

Hence, Lim x→ 0 (Sin 4x / x) = 4

Example

Calculate the Lim x→5 (x2 – 25) / (x – 5)

Solution:

Lim x→5 (x2 – 25) / (x – 5) = (25 – 25) / 5 – 5 = 0 / 0

This is an indeterminate form. We will solve it by using an algebraic technique.

= lim x→5 (x2 – 52) / (x – 5)

 ∴ (a2 – b2) = (a – b) (a + b)

= Lim x→5 (x – 5) (x + 5) / (x – 5)

After simplification, we get

= Lim x→5 (x + 5)

Apply the limit

= 5 + 5 = 10

So, the Lim x→5 (x2 – 25) / (x – 5) = 10

In this article, we have explored the limit, a fundamental concept of calculus. We have discussed the different types of limits. We covered the characteristics of limits in this article. We learned different methods to solve intermediate forms. We solved some examples for you to understand this topic in a better manner. After reading this article, you will be able to solve any complicated question of limits. But mathematics requires more than just practicing problems. Quizplus is a valuable resource for studying, as it not only offers practice questions but also provides comprehensive explanations and step-by-step solutions.
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