Limits & Functions | Properties, Algebra, Continuity, Solved Examples, FAQs, Practice Problems

Limits – Class 11 Mathematics

Written by Neeraj Anand | ANAND CLASSES
Published by ANAND TECHNICAL PUBLISHERS

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🔍 Introduction to Limits

In calculus, limits form the foundation for understanding concepts like continuity, derivatives, and integrals. A limit defines the behavior of a function as the input approaches a particular value. In simple terms, it answers the question:

“What value does a function approach as the input gets closer to a specific point?”

Limits Definition

Let us consider a real-valued function “f” and the real number “a”, 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”. The “lim” shows limit, and fact that function f(x) approaches the limit L as x approaches a is described by the right arrow.

One-Sided Limits

There are two path to approach any point in 2D space along a curve. That are from Left Hand Side of Curve or Right Hand Side of Curve. Approaching the curve from either sides allow us to find two separate limit 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 llim_{x→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→a+ f(x)=L.

Two-Sided Limits

Two-sided limits, also known as bilateral limits 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=a, except possibly at x=a itself. The two-sided limit of f(x) as x approaches a, denoted as:

lim_x→a​ f(x)

exists if and only if both the left-hand limit (as x approaches a from the left) and the right-hand limit (as x approaches a 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. If the value of f(x) becomes arbitrarily large (positive or negative) as x approaches a certain value a, the limit is said to be infinite.

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

The following are the properties of limits.

\(\begin{array}{l}\text{We assume that }\lim\limits_{x \to a}f(x)\text{ and }\lim\limits_{x \to a} g(x) \text{ exist and c is a constant. Then,}\end{array} \)

• \(\begin{array}{l}\lim\limits_{x \to a} \left [ c.f(x) \right ]= c \lim\limits_{x \to a}f(x)\end{array} \)

• \(\begin{array}{l}\lim\limits_{x \to a} \left [ f(x)\pm g(x) \right ]= \lim\limits_{x \to a}f(x) \pm \lim\limits_{x \to a}g(x)\end{array} \)

• \(\begin{array}{l}\lim\limits_{x \to a} \left [ f(x) .g(x) \right ]= \lim\limits_{x \to a}f(x) . \lim\limits_{x \to a}g(x)\end{array} \)  

• \(\begin{array}{l}\large \lim\limits_{x \to a} \left [ \frac{f(x)}{g(x)} \right ] = \frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)} \text{ provided }\lim\limits_{x \to a}g(x) \neq 0\end{array} \)
• \(\begin{array}{l}\lim\limits_{x \to a} c = c\end{array} \)  
• \(\begin{array}{l}\lim\limits_{x \to a} xn = an\end{array} \)  

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

Algebra of Limits

Algebra of the limit of the function are added below,

Law of Addition	limx⇢a {f(x) + g(x)} = limx⇢a f(x) + limx⇢a g(x)
Law of Subtraction	limx⇢a {f(x) – g(x)} = limx⇢a f(x) – limx⇢a g(x)
Law of Multiplication	limx⇢a {f(x) . g(x)} = limx⇢a f(x) . limx⇢a g(x)
Law of Division	limx⇢a {f(x) / g(x)} = limx⇢a f(x) / limx⇢a g(x)

Limits and Functions

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

Limit of 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

Solved Examples on Limits

Example 1: 

\(\begin{array}{l}\text{To Compute } \mathbf{\lim \limits_{x \to -4} (5x^{2} + 8x – 3)}\end{array} \)

Solution:

First, use property 2 to divide the limit into three separate limits. Then use property 1 to bring the constants out of the first two. This gives,

\(\begin{array}{l}\lim\limits_{x \to -4} (5x^{2}+8x-3) = \lim\limits_{x \to -4} (5x^{2})+ \lim\limits_{x \to -4} (8x)- \lim\limits_{x \to -4} (3)\end{array} \)

\(\begin{array}{l}=5(-4)^{2}+ 8(-4)- 3 \end{array} \)

\(\begin{array}{l}= 80 – 32 – 3 =45\end{array} \)

Example 2: 

\(\begin{array}{l}\text{To Compute }\mathbf{\lim\limits_{x \to 6} \left [ \frac{(x-3)(x-2)}{x-4} \right ]}\end{array} \)

Solution: 

\(\begin{array}{l}\text{Given }\lim\limits_{x \to 6} \left [ \frac{(x-3)(x-2)}{x-4} \right ]\end{array} \)

\(\begin{array}{l}= \left [ \frac{ \lim\limits_{x \to 6} (x-3) \lim\limits_{x \to 6}(x-2)}{ \lim\limits_{x \to 6}(x-4)} \right ]\end{array} \)

\(\begin{array}{l}= \left [ \frac{ (6-3) (6-2)}{ (6-4)} \right ]\end{array} \)

\(\begin{array}{l}= \left [ \frac{ (3) (4)}{ (2)} \right ] =6\end{array} \)

Example 3:

\(\begin{array}{l}\text{Compute }\mathbf{\lim \limits_{x \to 3} \frac{(x^{2}- 9)}{x – 3}}\end{array} \)

Solution:

\(\begin{array}{l}\text{Given }\lim \limits_{x \to 3} \frac{(x^{2}- 9)}{x – 3}\end{array} \)

It is to be noted that, on substituting the value 3 directly to the funciton, the nemerator as well as denominator will become 0, and we know the value 0/0, does not exist.

Using the property of squares , we have:

\(\begin{array}{l}\lim \limits_{x \to 3} \frac{(x-3)(x+3)}{x – 3} = \lim \limits_{x \to 3}(x+3)\end{array} \)

= 6

Example 4: lim _{x ⇢ 6} x/3 
Solution:

lim _{x ⇢ 6} x/3 = 6/3 = 2

Example 5: 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 6: 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

FAQs on Limits in Calculus

What is Limit in Calculus?

A limit in calculus is defined as a function that approaches a particular value as the independent variable of the function approaches a particular value.

What is Limit Formula?

Limit formula is the formula that is used to calculate the limit of the given function. Let’s take a function y = f(x) and we take a point x = a then limit of function f(x) is defined as,

Iim_x→0 f(x) = f(a)

What is Limit of a Function Class 11?

A limit of a function f(x) is defined as a value, where the function reaches as the limit reaches some value.

When can Limit Not Exist?

If the Left Hand Limit and the right hand limit of the function are not equal then we say that limit of the function does to exist, i.e. limit does not exit if,

LHL ≠ RHL

How to Find Limit in Calculus?

Limit of the function is found by substituting the value of the function that the limit approaches if the limit exist. When the limit of the function does not exist then we first simplify the function and then find the limit of the function.

What is the use of Limit?

The limit of the function is used for various purposes that are,

• Limits are used to find the limiting value of the function.
• Limits are used to find the derivative of the function.
• Limits are also used to define the integral value of the function, etc.

What are Three Types of Limits?

The three types of limits are,

• Two-Sided Limits
• One-Sided Limits
• Infinite Limits

📝 Methods to Solve Limits

1. Direct Substitution – Substitute the value directly into the function.
2. Factorization – Simplify expressions by factoring and canceling terms.
3. Rationalization – Multiply by conjugate expressions to simplify square roots.
4. L’Hôpital’s Rule – Apply derivatives to resolve indeterminate forms like 00\frac{0}{0}00​ or ∞∞\frac{\infty}{\infty}∞∞​.

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📚 Real-Life Applications of Limits

• Determining instantaneous speed in physics.
• Calculating slopes of curves using derivatives.
• Analyzing population growth models in biology and economics.

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