Limits & Continuity

Applied Mathematics Class 11 - Chapter 7

1. Introduction to Limits

Concept of Limit

The limit of a function $f(x)$ as $x$ approaches $a$ is the value $L$ that $f(x)$ gets closer to as $x$ gets closer to $a$ (from both sides). Written as $\lim_{x \to a} f(x) = L$ .

"We care about what happens NEAR $x=a$, not AT $x=a$."

Left & Right Hand Limits

Left Hand Limit (LHL): $\lim_{x \to a^-} f(x)$ and Right Hand Limit (RHL): $\lim_{x \to a^+} f(x)$ . Limit exists if LHL = RHL.

Visualizing the Limit

Consider

f(x) = \frac{x^2 - 1}{x - 1}

. At

x=1

, it is undefined (

0/0

).
Use the mouse to approach

x=1

from both sides.

Left Hand Limit:1.99...

→ 2 ←

Right Hand Limit:2.01...

2. Algebra & Standard Results

Algebra of Limits

$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ , provided $\lim_{x \to a} g(x) \neq 0$

Standard Algebraic Limit

\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}

Very useful for polynomial fractions.

Exp & Log Limits

$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
$\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

3. Methods of Evaluation

⚠️

Indeterminate Forms

Watch out for: Forms like

\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \times \infty

where the limit cannot be determined by direct substitution.

Direct Substitution

Just plug in the value of $a$ into the function.

Ex:

\lim_{x \to 1} (x^2 + 3)

1.Replace $x$ with 1: $1^2 + 3$ .
2.Evaluate: $1+3 = 4$ .
3.Since we didn't get $0/0$ , this is the answer.

Factorization

Used when you get $0/0$ . Factor numerator and denominator to cancel common terms.

Ex:

\lim_{x \to 2} \frac{x^2-4}{x-2}

1.Plug in 2: $(4-4)/(2-2) = 0/0$ (Indeterminate).
2.Factor numerator: $(x-2)(x+2)$ .
3.Cancel $(x-2)$ : $\lim_{x \to 2} (x+2)$ .
4.Substitute 2: $2+2=4$ .

Rationalization

Used for square roots giving $0/0$ . Multiply by conjugate.

Ex:

\lim_{x \to 0} \frac{\sqrt{1+x}-1}{x}

1.Multiply Num & Denom by $\sqrt{1+x}+1$ .
2.Numerator becomes $(1+x)-1 = x$ .
3.Cancel $x$ : $\lim_{x \to 0} \frac{1}{\sqrt{1+x}+1}$ .
4.Substitute 0: $1/(1+1) = 1/2$ .

4. Continuity

Definition

A function $f(x)$ is continuous at $x=c$ if: 1. $f(c)$ is defined, 2. $\lim_{x \to c} f(x)$ exists, and 3. $\lim_{x \to c} f(x) = f(c)$ .

Intuitively: A graph you can draw without lifting your pen.

ℹ️

Discontinuity Types

Types: Removable (Hole), Jump, Infinite.

Continuity vs Discontinuity

Status: Continuous. You can draw this graph without lifting your pen. LHL = RHL = f(c).

Important Concept Graphs

Modulus Function

f(x) = |x|

modulus

Graph of Modulus Function

Domain

$\mathbb{R}$ (All Real Numbers)

Continuity Status

Continuous everywhere. Sharp corner at

x=0

means not differentiable, but still continuous (no break).

Greatest Integer Function

f(x) = [x]

step

Graph of Greatest Integer Function

Domain

$\mathbb{R}$

Continuity Status

Discontinuous at every integer. **Critical Note:** For any non-integer

x

[x]

shifts the value to the **LEFT (lower)** integer. Example:

[2.3] = 2

and

[-1.5] = -2

. (If it shifted right, it would be the Ceiling function).

Reciprocal Function

f(x) = 1/x

reciprocal

Graph of Reciprocal Function

Domain

$\mathbb{R} - \{0\}$ (Because division by 0 is undefined)

Continuity Status

Discontinuous at

x=0

(Infinite Discontinuity). The graph breaks into two separate branches because

1/0

is undefined.

Signum Function

f(x) = |x|/x

signum

Graph of Signum Function

Domain

$\mathbb{R} - \{0\}$

Continuity Status

Discontinuous at

x=0

(Jump Discontinuity). Left Limit = -1, Right Limit = 1.

Slope with Jump

f(x) = x + ext{sgn}(x)

slope_jump

Graph of Slope with Jump

Domain

$\mathbb{R} - \{0\}$

Continuity Status

Discontinuous at

x=0

. As

x o 0^-

y o -1

. As

x o 0^+

y o 1

. Shows limits exist but are unequal.

Loading Knowledge...

Preparing your next chapter

1. Introduction to Limits

Concept of Limit

Left & Right Hand Limits

Visualizing the Limit

2. Algebra & Standard Results

Algebra of Limits

Standard Algebraic Limit

Exp & Log Limits

3. Methods of Evaluation

Indeterminate Forms

Direct Substitution

Factorization

Rationalization

4. Continuity

Definition

Discontinuity Types

Continuity vs Discontinuity

Important Concept Graphs

Modulus Function

Graph of Modulus Function

Domain

Continuity Status

Greatest Integer Function

Graph of Greatest Integer Function

Domain

Continuity Status

Reciprocal Function

Graph of Reciprocal Function

Domain

Continuity Status

Signum Function

Graph of Signum Function

Domain

Continuity Status

Slope with Jump

Graph of Slope with Jump

Domain

Continuity Status

📝 Practice Zone

Concept Check Problems

🔥 CBSE Exam Priority Questions

Important Questions (Applied Maths)

Loading Knowledge...

1. Introduction to Limits

Concept of Limit

Left & Right Hand Limits

Visualizing the Limit

2. Algebra & Standard Results

Algebra of Limits

Standard Algebraic Limit

Exp & Log Limits

3. Methods of Evaluation

Indeterminate Forms

Direct Substitution

Factorization

Rationalization

4. Continuity

Definition

Discontinuity Types

Continuity vs Discontinuity

Important Concept Graphs

Modulus Function

Graph of Modulus Function

Domain

Continuity Status

Greatest Integer Function

Graph of Greatest Integer Function

Domain

Continuity Status

Reciprocal Function

Graph of Reciprocal Function

Domain

Continuity Status

Signum Function

Graph of Signum Function

Domain

Continuity Status

Slope with Jump

Graph of Slope with Jump

Domain