Limits in Calculus

In mathematics, a limit is a fundamental concept that describes the behavior of a function or sequence as its input approaches a particular value.

Limits are used in calculus to define derivatives, continuity, and integrals, and they represent the value that a function approaches as the input approaches a certain value.

• Let's say we have a function f(x) = x². In the graph given above, notice that as x⇢0, f(x) also tends to become zero.
• This can be written in terms of a limit as \bold{\lim_{x \to 0} f(x) = 0} . It is read as the limit of f(x) as x tends to zero. 

In general, as x ⇢ a, f(x) ⇢ l, then l is called the limit of the function f(x). It can also be written as, \bold{\lim_{x \to a}f(x) = l}

Sometimes some functions are not continuous. That is, they appear to be approaching two different values when they are approached from two sides. For example, let's see this step function given in the figure below. 

This function can be defined as, 

f(x)= \begin{cases} 1,& \text{if } x > 0\\ 0,& \text{if } x = 0\\ -1,& \text{if x < 0}\end{cases}

Suppose we want to find the limit of this function as x approaches zero. This naturally leads to directions from which we can approach. Left-hand side and the right-hand side limits.

The right-hand side limit is the value of the function that it takes while approaching it from the right-hand side of the desired point. Similarly, the left-hand side limit is the value of function while approaching it from the left-hand side. 

For this particular function,

Left-hand side limit, lim_x→0^- f(x) = -1

Right-hand side limit, lim_x→0⁺ f(x) = 1

Mathematical Expression for Limit

To define the limit of a function let us consider a real-valued function “f” and the real number “a” such that the variable of the function approaches the value "a" then the limit is normally defined as:

lim_x⇢a f(x) = L

It is read as “the limit of f of x, as x approaches a equals L”.

For any function f(x) defined for all x ≠ a over an open interval containing a. Now suppose we have a real number L such that,

lim_x⇢a f(x) = L

Then for every ε > 0, there exists a δ > 0, such that, 0 < |x - a| < δ the,

|f(x) - L| < ε

Types of Limits

Limits in math are of several types, each describing different situations and behaviors of functions as the independent variable approaches a certain value or infinity. Here are the main types of limits:

One-Sided Limits

There are two paths to approach any point in 2D space along a curve. That are from Left Hand Side of the Curve or Right Hand Side of the Curve. Approaching the curve from either sides allow us to find two separate limits of the function. These two limits are called:

• Left Hand Limit (LHL): The limit as the variable approaches the value from the left side. It is represented as \lim_{x \to a^-} f(x) = L.
• Right Hand Limit (RHL): The limit as the variable approaches the value from the right side. It is represented as \lim_{x \to a^+} f(x) = L.

Two-Sided Limits

Two-sided limits, also known as bilateral limits, are a fundamental concept in calculus that describe the behavior of a function as the independent variable approaches a particular value from both the left and the right sides simultaneously.

Formally, let f(x) be a function defined on an open interval containing x=c, except possibly at x=c itself. The two-sided limit of f(x) as x approaches c, denoted as:

lim_x→c​ f(x)

exists if and only if both the left-hand limit (as x approaches c from the left) and the right-hand limit (as x approaches c from the right) exist and are equal.

Infinite Limits

Infinite limits occur when the value of a function approaches positive or negative infinity as the independent variable approaches a particular point. Formally, if the value of f(x) becomes arbitrarily large (positive or negative) as x approaches a certain value c, the limit is said to be infinite.

• Positive Infinite Limit: If f(x) increases without bound as x approaches c, the limit is denoted as lim_x→c ​f(x) = +∞.
• Negative Infinite Limit: If f(x) decreases without bound as x approaches c, the limit is denoted as lim_x→c​f(x) = −∞.

For instance, consider the function f(x) = 1/x²​. As x approaches 0 from either the positive or negative direction, f(x) becomes increasingly large (approaches infinity), so the limit of f(x) as x approaches 0 is +∞.

Limits at Infinity

Limit at infinity describe the behavior of a function as the independent variable grows without bound (approaches positive or negative infinity).

• Limit at Positive Infinity: If f(x) approaches a finite limit as x goes to positive infinity, it is denoted as lim_x→+∞ ​f(x) = L.
• Limit at Negative Infinity: If f(x) approaches a finite limit as x goes to negative infinity, it is denoted as lim_x→−∞ ​f(x) = L.

For example, consider the function f(x) = 1/x​. As x grows without bound (either positively or negatively), f(x) approaches 0. Thus, lim_x→+∞ ​1/x ​ = 0 and lim_x→−∞ 1/​x ​= 0.

Properties of Limits

Various properties of the limit of a function are,

• lim _{x ⇢ a}  k = k, where k is a constant quantity
• The value of lim _{x ⇢  a} x = a
• Value of lim _{x ⇢ a} bx + c = ba + c
• lim _{x ⇢ a} xⁿ = aⁿ if n is a positive integer.
• Value of lim _{x  ⇢ +0} 1/x^r  = +∞  
• lim _{x ⇢ −0} 1/x^r  = −∞, if r is odd
• lim _{x ⇢ −0} 1/x^r  = +∞, if r is even

Algebra of Limit

Algebra of the limit of the function are added below,

Special Rules of Limit

Various rules that are used to simplify the limit of the function are,

• lim_x⇢a (xⁿ - aⁿ)/(x - a) = na^(n-1)
• lim_x⇢a sin x/x = 1
• lim_x⇢a tan x/x = 1
• lim_x⇢a (1 - cos x)/x = 0
• lim_x⇢a cos x = 1
• lim_x⇢a e^x = 1
• lim_x⇢a (e^x - 1)/x = 1
• lim_x⇢∞ (1 + 1/x)^x = e

Note:

• The limits involving trigonometric and exponential functions (such as \frac{\sin x}{x}, \frac{\tan x}{x},\frac{1 - \cos x}{x}, and \ \frac{e^x - 1}{x} )are valid only as x→0 .
• The formula \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1} holds for any real value of a.
• The limit \lim_{x \to \infty} (1 + \frac{1}{x})^x = e defines the mathematical constant e.

Also Check:

• How to Find Limits?
• One-Sided Limits
• Formal Definition of Limits

Limits and Functions

Limit of any function is defined as the value that the function approaches when the independent variable of the function approaches a particular value. A function's limit exists only when the left hand limit and right hand limit of the function both exist and are equal.

Limit of a Polynomial Function

Limit of the polynomial function are added below, consider a polynomial function,

f(x) = a₀ + a₁x + a₂x² + … + a_nxⁿ

Here, a₀, a₁, … , a_n are all constants. At any point x = a, the limit of this polynomial function is

lim_x⇢a f(x) = lim _x⇢a [a₀ + a₁x + a₂x² + . . .  + a_nxⁿ]
= lim_x⇢a a₀ + a₁lim _{x ⇢ a} x + a₂lim _{x ⇢ a} x² + . . . + a_nlim _{x ⇢ a} xⁿ
= lim_x⇢a  a₀ + a₁a + a₂a² + . . . + a_naⁿ
= lim_x⇢a = f(a)

Limit of Rational Function

The limit of any rational function of the type m(x)/n(x), where n(x) ≠ 0 and m(x) and n(x) are polynomial functions, is:

lim_x⇢a [m(x)/n(x)]
= lim_x⇢a m(x)/lim_x⇢a n(x)
= m(a)/m(b)          

The very first step to find the limit of a rational function is to check if it is reduced to the form 0/0 at some point. If it is so, then we need to do some adjustments so that one can calculate the value of the limit. This can be done by canceling the factor which causes the limit to be of the form 0/0. For example,

f(x) = (x² - 4x + 4)/(x² − 4)

Taking limit over it for x = 2, the function is of the form 0/0,

lim _{x ⇢ 2} f(x)
= lim _{x ⇢ 2} (x² - 4x + 4)/(x² − 4)
= lim _{x ⇢ 2} [( x - 2)²/(x + 2)( x - 2)]
= lim _{x ⇢ 2} [(x - 2)/(x + 2)]
= 0/4 ( ≠ 0/0 ) = 0

Limits of Complex Functions

If we are given a complex function then the limit of the complex function is calculated as, suppose we are given a function f(z) where z is a complex variable then the z = z₀ then the f(z) is differentiable if,

lim_Δz→0 [f(z₀ + Δz) - f(z₀)]/Δz

Where, Δz = Δx + iΔy

Limits of Exponential Functions

The limit of exponential function is easily calculated by taking into consideration the initial value of the exponential function. Suppose we are given an exponential function f(x) = a^x where a > 0.

For f(b) > 1

• lim_x→∞ a^x = ∞
• lim_x→-∞ a^x = 0

For 0 < f(b) < 1

• lim_x→∞ a^x = 0
• lim_x→-∞ a^x = ∞

Limit of a Function of Two Variables

For the given function with two variables say f(x, y) then suppose if the limit of the function is C, (x, y) → (a, b) provided that ϵ > 0 here exists Δ > 0 such that |f(x, y) - C| < ϵ whenever 0 < √{(x -a)² + (y - b)²} < Δ. Then,

Iim _{(x, y) → (a, b)} f(x, y) = C

Related Article:

• Limit Formulas
• Limits by Direct Substitution
• Squeeze Theorem
• L'Hopital's Rule

Solved Examples of Limits

Example 1: lim_x⇢6 x/3 
Solution:

lim_x⇢6 x/3 = 6/3 = 2

Example 2: lim_x⇢2 (x² - 4)/(x - 2)
Solution:

As we know, (x² - 4) = (x² - 2²) = ( x - 2 )( x - 2 ) 

Now, lim_x⇢2 (x² - 4)/(x - 2)
= lim_x⇢2 (x- 2)(x + 2)/(x - 2)
= lim_x⇢2 (x + 2)
= 4

Example 3: lim_x⇢1/2 (2x - 1)/(4x² - 1)
Solution:

As we know, 4x² - 1 = (2x²) - (1²) = (2x + 1) (2x - 1)

Now, lim_x⇢1/2  (2x - 1)/(4x² - 1)
= lim_x⇢1/2  (2x- 1)/(2x - 1) (2x + 1)
= lim_{x⇢1/ 2} 1/(2x + 1)
= 1/{2 × (1/2) + 1} = 1/2

Example 4: Find the right-handed limit: \lim_{x \to 0^+} f(x) \quad \text{where} \quad f(x) = \begin{cases} 1 & \text{if } x > 0 \\0 & \text{if } x = 0 \\-1 & \text{if } x < 0\end{cases}
Solution:

Since we're asked to find the right-hand limit (x→0⁺), we are interested in the value of the function as xxx approaches 0 from the positive side (i.e., from values greater than 0).

• For x > 0, the function f(x) = 1
• Thus, the right-hand limit is:

\lim_{x \to 0^+} f(x) = 1

Example 5: lim_x→0 (1/x)
Solution:

We need to evaluate the limit of 1/x​ as x approaches 0 from the left-hand side (i.e., as x approaches 0 from negative values).

• As x approaches 0 from the left (i.e., for negative values of x), 1/x becomes increasingly negative.
• Thus, the limit is:

lim⁡_x→0 1/x = −∞

Example 6: Evaluate the two-sided limit: \lim_{x \to 0} f(x) \quad \text{where} \quad f(x) = \begin{cases} 1 & \text{if } x > 0 \\-1 & \text{if } x < 0\end{cases}
Solution:

In this case, we need to check both the left-hand and right-hand limits to determine the two-sided limit.

• For x > 0, f(x) = 1
• For x < 0, f(x) = −1

Thus, we have:

• Right-hand limit: lim⁡_x→0 f(x) = 1
• Left-hand limit: lim⁡_x→0 f(x) = -1

Since the left-hand limit does not equal the right-hand limit, the two-sided limit does not exist:

lim⁡_x→0 f(x)does not exist.

Example 7: Find the limit: lim_x→0(1/x²)
Solution:

We need to find the limit of 1/x²​ as x approaches 0.

• As x approaches 0 (from either the left or the right), x² gets smaller and smaller, and thus 1/x² becomes larger and larger without bound.
• Therefore, the limit is:

lim_x→0 (1/x²) = +∞.

Example 8: lim_x→4 (x² - 16)/(x - 4)
Solution:

First, notice that x² − 16 is a difference of squares, so we can factor it: x² − 16 = (x − 4)(x + 4)

Now, the limit becomes: \lim_{x \to 4} \frac{(x - 4)(x + 4)}{x - 4}

Cancel out the (x−4) terms in the numerator and denominator (for x≠4):

lim_⁡x→4(x + 4):

Substitute x = 4:
4 + 4 = 8

Thus, the limit is:

\lim_{x \to 4} \frac{x^2 - 16}{x - 4} = 8

Practice Questions on Limits

1. lim_x→-2 (x² - 1)/(x + 2)
2. lim_x→3 (x³ - 27)/(x - 3)
3. lim_x→5 (x² + 2x - 15)/(x - 5)
4. Find the right-handed limit: lim_x→2 (x² - 4)/(x - 2)
5. Find the limit: lim_x→0 (1/x³)
6. Evaluate the two-sided limit: lim_x→0 f(x) where f(x) = { 1 if x > 0, -1 if x < 0 }
7. Find the limit: lim_x→0 (1/x²)
8. Find the limit: lim_⁡x→3 (x²−9/x−3)