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Limits of Functions

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Limits of Functions

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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