Limits of Functions

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# Limits of Functions ## 🎯 Learning Objectives By the end of this topic, you will be able to: 1. **Define** the limit of a function at a point 2.

Limits of Functions

🎯 Learning Objectives

By the end of this topic, you will be able to:

Define the limit of a function at a point
Compute one-sided limits and determine when a two-sided limit exists
Evaluate limits using algebraic manipulation, factoring, rationalization
Apply standard limit results: $\frac{\sin x}{x}$ , $\frac{e^x-1}{x}$ , $\frac{\ln(1+x)}{x}$
Recognize and resolve indeterminate forms using L'Hôpital's rule

📋 Prerequisites

Sequences — concept of convergence
Functions — function behavior near a point
Polynomials — factoring techniques

📖 Core Content

11.1 Intuition: Approaching a Point

What happens to

f(x) = x^2

x

gets closer and closer to

3

? The output approaches

9

. We write

\lim_{x \to 3} x^2 = 9

. The limit describes the trend of

f(x)

near

x = a

, regardless of whether

f(a)

is defined. This is crucial for understanding derivatives and continuity.

💡 Why this matters: Limits are the foundation of calculus. Derivatives = limits of difference quotients. Integrals = limits of Riemann sums. Everything in calculus rests on limits.

11.2 Definition

\lim_{x \to a} f(x) = L

Means: as

x

gets arbitrarily close to

a

(from either side),

f(x)

gets arbitrarily close to

L

11.2.1 One-Sided Limits

Left-hand limit: $\lim_{x \to a^-} f(x)$ — approach $a$ from below (smaller values)
Right-hand limit: $\lim_{x \to a^+} f(x)$ — approach $a$ from above (larger values) Two-sided limit exists iff both one-sided limits exist and are equal:

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

(Diagram)

11.3 Properties of Limits

\lim_{x \to a} f(x) = L

and

\lim_{x \to a} g(x) = M

Property	Formula
Sum	$\lim[f(x) + g(x)] = L + M$
Difference	$\lim[f(x) - g(x)] = L - M$
Product	$\lim[f(x) \cdot g(x)] = L \cdot M$
Quotient	$\lim f(x)/g(x) = L/M$ (if $M \neq 0$ )
Constant multiple	$\lim c \cdot f(x) = cL$
Power	$\lim [f(x)]^n = L^n$
Root	$\lim \sqrt[n]{f(x)} = \sqrt[n]{L}$ (if $L \geq 0$ for even $n$ )

11.4 Standard Limits (Very Important!)

Formula	Notes
$\displaystyle \lim_{x \to 0} \frac{\sin x}{x} = 1$	$x$ in radians
$\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} = 0$
$\displaystyle \lim_{x \to 0} \frac{e^x - 1}{x} = 1$
$\displaystyle \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$
$\displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$
$\displaystyle \lim_{x \to 0} (1 + x)^{1/x} = e$	Same as above

11.5 Evaluation Techniques

11.5.1 Direct Substitution

f

is continuous at

a

\lim_{x \to a} f(x) = f(a)

. Works for polynomials, rational functions (where denominator ≠ 0), trig functions, exponentials.

11.5.2 Factoring

When substitution gives

0/0

, factor and cancel. Example:

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

\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4

11.5.3 Rationalization

When radicals cause

0/0

, multiply by conjugate. Example:

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

\lim_{x \to 0} \frac{(\sqrt{x+1}-1)(\sqrt{x+1}+1)}{x(\sqrt{x+1}+1)} = \lim_{x \to 0} \frac{x}{x(\sqrt{x+1}+1)} = \lim_{x \to 0} \frac{1}{\sqrt{x+1}+1} = \frac{1}{2}

11.5.4 Limits at Infinity

Divide numerator and denominator by highest power of

x

. Example:

\lim_{x \to \infty} \frac{3x^2 + 2x}{4x^2 - 1} = \frac{3}{4}

11.6 Indeterminate Forms

Form	Example
$0/0$	$\frac{x^2-4}{x-2}$ at $x=2$
$\infty/\infty$	$\frac{x^2+1}{x-1}$ as $x\to\infty$
$0 \cdot \infty$	$x \cdot \ln x$ as $x\to 0^+$
$\infty - \infty$	$\sqrt{x+1} - \sqrt{x}$ as $x\to\infty$
$0^0$	$x^x$ as $x\to 0^+$
$1^\infty$	$(1+1/x)^x$ as $x\to\infty$
$\infty^0$	$x^{1/x}$ as $x\to\infty$

11.7 L'Hôpital's Rule

\lim_{x \to a} f(x)/g(x)

0/0

\infty/\infty

, and

\lim f'(x)/g'(x)

exists:

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Important: Only applies to

0/0

\infty/\infty

forms. Must differentiate numerator and denominator separately (not quotient rule!).

Worked Examples

Example 4.1:

\lim_{x \to 0} \frac{\sin x}{x}

(using L'Hôpital)

\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1

Example 4.2:

\lim_{x \to \infty} \frac{x^2}{e^x}

\lim_{x \to \infty} \frac{x^2}{e^x} = \lim_{x \to \infty} \frac{2x}{e^x} = \lim_{x \to \infty} \frac{2}{e^x} = 0

Example 4.3:

\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}

\lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \lim_{x \to 0} \frac{e^x - 1}{2x} = \lim_{x \to 0} \frac{e^x}{2} = \frac{1}{2}

📐 Key Formulas — Summary Table

Form	Technique
$0/0$ (polynomial)	Factor and cancel
$0/0$ (radical)	Rationalize
$\infty/\infty$	Divide by highest power
$0/0$ or $\infty/\infty$	L'Hôpital's rule
$0 \cdot \infty$	Rewrite as $0/0$ or $\infty/\infty$
$\infty - \infty$	Common denominator or conjugate
$1^\infty$	Use $\lim = e^{\lim (f(x)-1)g(x)}$

\boxed{\text{Standard limit: } \lim_{x\to 0} \frac{\sin x}{x} = 1,\ \lim_{x\to 0} \frac{e^x-1}{x} = 1,\ \lim_{x\to 0} \frac{\ln(1+x)}{x} = 1}

⚠️ Common Pitfalls

Pitfall 1: Applying L'Hôpital to Non-Indeterminate Forms

\lim_{x \to 1} \frac{x^2}{x} = 1

directly. Using L'Hôpital:

\frac{2x}{1} = 2

— WRONG! The form was

1/1

, not

0/0

\infty/\infty

Pitfall 2: Confusing One-Sided Limits with Two-Sided

\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)

, then

\lim_{x \to a} f(x)

does NOT exist.

Pitfall 3: Forgetting That $\sin x/x$ Requires Radians

The limit

\lim_{x\to 0} \frac{\sin x}{x} = 1

only holds when

x

is in radians.

📝 Practice Questions

Q1: Find $\lim_{x \to 3} (2x + 1)$ .

Direct substitution: $2(3) + 1 = 7$

$\boxed{7}$ Q2: Find $\lim_{x \to 2} \frac{x^2 - 3x + 2}{x-2}$ .

Factor: $\frac{(x-1)(x-2)}{x-2} = x-1 \to 1$

$\boxed{1}$ Q3: Find $\lim_{x \to 0} \frac{\sqrt{4+x} - 2}{x}$ .

Rationalize: $\frac{(\sqrt{4+x} - 2)(\sqrt{4+x}+2)}{x(\sqrt{4+x}+2)} = \frac{x}{x(\sqrt{4+x}+2)} = \frac{1}{\sqrt{4+x}+2} \to \frac{1}{4}$

$\boxed{\frac{1}{4}}$ Q4: Find $\lim_{x \to \infty} \frac{5x^2 - 3x + 1}{2x^2 + 7}$ .

Divide by $x^2$ : $\frac{5 - 3/x + 1/x^2}{2 + 7/x^2} \to \frac{5}{2}$

$\boxed{\frac{5}{2}}$ Q5: Find $\lim_{x \to 0} \frac{\sin 3x}{x}$ .

$\frac{\sin 3x}{x} = 3 \cdot \frac{\sin 3x}{3x} \to 3 \cdot 1 = 3$

$\boxed{3}$ Q6: Find $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$ .

$= \lim_{x\to 0} \frac{\sin x}{2x} = \frac{1}{2} \cdot 1 = \frac{1}{2}$

$\boxed{\frac{1}{2}}$ Q7: Find $\lim_{x \to 0} \frac{e^{2x} - 1}{x}$ .

$= 2 \cdot \lim_{x\to 0} \frac{e^{2x} - 1}{2x} = 2 \cdot 1 = 2$

$\boxed{2}$ Q8: Evaluate $\lim_{x \to 0} \frac{x^2}{\sin x}$ .

$= \lim_{x\to 0} \frac{x}{\sin x} \cdot x = 1 \cdot 0 = 0$

$\boxed{0}$ Q9: Find $\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$ .

Factor: $\frac{(x-1)(x^2+x+1)}{x-1} = x^2 + x + 1 \to 3$

$\boxed{3}$ Q10: Find $\lim_{x \to 0^+} x \ln x$ .

Rewrite as $\frac{\ln x}{1/x}$ : $-\infty/\infty$ form. L'Hôpital: $\frac{1/x}{-1/x^2} = -x \to 0$

$\boxed{0}$ Q11: Find $\lim_{x \to 0} \frac{e^x - 1}{\sin x}$ .

L'Hôpital: $\frac{e^x}{\cos x} \to \frac{1}{1} = 1$

$\boxed{1}$ Q12: Find $\lim_{x \to 0} (1 + 2x)^{1/x}$ .

This is $1^\infty$ form. $\lim_{x\to 0} (1+2x)^{1/x} = e^{\lim_{x\to 0} \frac{\ln(1+2x)}{x}} = e^{\lim_{x\to 0} \frac{2}{1+2x}} = e^{2}$

$\boxed{e^2}$

🔗 Cross-References

Previous topic: Sequences
Next topic: Continuity
Related: Derivatives — derivatives defined via limits
Across courses: BSMA1003 Maths 2 (multivariable limits, continuity) Join Discord Previous7.1 Sequences & ConvergenceNext7.3 Continuity

What this note is really saying

Limits of Functions

🎯 Learning Objectives

📋 Prerequisites

📖 Core Content

11.1 Intuition: Approaching a Point

11.2 Definition

11.2.1 One-Sided Limits

11.3 Properties of Limits

11.4 Standard Limits (Very Important!)

11.5 Evaluation Techniques

11.5.1 Direct Substitution

11.5.2 Factoring

11.5.3 Rationalization

11.5.4 Limits at Infinity

11.6 Indeterminate Forms

11.7 L'Hôpital's Rule

Worked Examples

📐 Key Formulas — Summary Table

⚠️ Common Pitfalls

Pitfall 1: Applying L'Hôpital to Non-Indeterminate Forms

Pitfall 2: Confusing One-Sided Limits with Two-Sided

Pitfall 3: Forgetting That sin⁡x/x\sin x/xsinx/x Requires Radians

📝 Practice Questions

🔗 Cross-References

Intelligence Hub

Pitfall 3: Forgetting That $\sin x/x$ Requires Radians