MA221
Week 2 Complete Mastery Guide
Continuity, Taylor Expansion & L’Hôpital’s Rule
HIGH EXAM WEIGHT TOPIC!
↭ Detailed Theory & Rigorous Definitions
↭ Continuity (ω-ε) & Properties
↭ Taylor’s Theorem & Approximations
↭ L’Hôpital’s Rule for Indeterminate Forms
↭ Exam-Optimized Formula Sheets
↭ Fully Worked Past Paper Questions
↭ Common Mistakes & Examiner Traps
↭ One-Page Quick Reference
Based on LSE Course Materials 2024–25
Course Notes Chapter 1
Lecture Slides Week 2
Past Exam Papers 2023–2025
London School of Economics
Department of Mathematics
Academic Year 2025–26
,MA221 Week 2 Complete Guide 1
Contents
1 Week 2: Continuity, Taylor Expansion & L’Hôpital’s Rule 2
1.1 A. Core Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 What is Continuity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Why Does Continuity Matter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 What is Taylor’s Theorem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 What is L’Hôpital’s Rule? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 B. Rigorous Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 C. Key Theorems (Exam-Critical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 D. Formula & Identity Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Continuity Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Taylor’s Theorem Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.3 L’Hôpital’s Rule Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 E. Exam-Style Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 F. Fully Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 G. Common Mistakes (Examiner Traps) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 H. One-Page Cheat Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Additional Practice Problems 21
2.1 Quick Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Solutions to Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
,MA221 Week 2 Complete Guide 2
1 Week 2: Continuity, Taylor Expansion & L’Hôpital’s Rule
1.1 A. Core Theory
1.1.1 What is Continuity?
Informally, a function is continuous if you can draw its graph without lifting your pen. More rigorously,
continuity means the limit equals the function value.
KEY INSIGHT
Continuity combines two concepts:
1. The limit lim f (x) exists
x→c
2. The function value f (c) exists
3. They are equal: lim f (x) = f (c)
x→c
No abrupt jumps, breaks, or holes!
1.1.2 Why Does Continuity Matter?
Continuous functions have special properties:
Intermediate Value Theorem (IVT): If f is continuous on [a, b], then f takes every value between
f (a) and f (b)
Extreme Value Theorem (EVT): Continuous functions on closed intervals attain maximum and
minimum values
We can swap limits and function evaluations
Integrals are well-defined (Week 3)
1.1.3 What is Taylor’s Theorem?
Taylor’s Theorem provides polynomial approximations to functions. Instead of working with compli-
cated functions like ex or sin(x), we can approximate them with polynomials!
Why polynomials?
Easy to evaluate (just arithmetic)
Easy to di!erentiate and integrate
Easy to analyze
Can approximate any smooth function arbitrarily well
INTUITION: Taylor Approximation
Think of Taylor polynomials as building increasingly accurate approximations:
0th degree: P0 (x) = f (c) (constant, just the function value)
1st degree: P1 (x) = f (c) + f ↑ (c)(x → c) (tangent line)
→→
2nd degree: P2 (x) = f (c) + f ↑ (c)(x → c) + f 2(c) (x → c)2 (parabola matching curvature)
Each degree adds more accuracy by matching another derivative!
, MA221 Week 2 Complete Guide 3
WHY TAYLOR WORKS: Approximating ex
The Taylor polynomial for ex about x = 0 is:
x2 x3 xn
Pn (x) = 1 + x + + + ··· +
2! 3! n!
Let’s see how accurate this is at x = 1 (approximating e ↑ 2.71828):
P0 (1) = 1 (o! by 63%)
P1 (1) = 2 (o! by 26%)
P2 (1) = 2.5 (o! by 8%)
P3 (1) = 2.667 (o! by 2%)
P5 (1) = 2.7167 (o! by 0.06%!)
Higher degree ↓ better approximation!
1.1.4 What is L’Hôpital’s Rule?
L’Hôpital’s Rule is a powerful tool for evaluating indeterminate forms like 00 or ↓
↓ . Instead of algebraic
manipulation, you di!erentiate the numerator and denominator separately!
Why it works: When both f (x) and g(x) approach 0, the ratio fg(x) (x)
behaves like the ratio of their
f → (x)
rates of change, which is g → (x) .
QUICK EXAMPLE: L’Hôpital in Action
sin(x)
Evaluate lim
x→0 x
Without L’Hôpital: Would need Taylor series or geometric arguments (messy!)
With L’Hôpital:
Check form: As x ↔ 0, both sin(x) ↔ 0 and x ↔ 0 ↓ Form is 0
0 ↭
sin(x) cos(x)
Apply rule: lim = lim =1
x→0 x x→0 1
Done! Much easier than other methods.
When NOT to use L’Hôpital:
When the form is NOT indeterminate (direct substitution works)
When algebraic simplification is easier (factoring, rationalizing)
When you’re asked to use a specific method (Taylor, definition, etc.)
1.2 B. Rigorous Definitions
Definition 2.1: Continuity at a Point
A function f is continuous at x = c if:
lim f (x) = f (c)
x→c
Equivalently, using the ω-ε definition:
Week 2 Complete Mastery Guide
Continuity, Taylor Expansion & L’Hôpital’s Rule
HIGH EXAM WEIGHT TOPIC!
↭ Detailed Theory & Rigorous Definitions
↭ Continuity (ω-ε) & Properties
↭ Taylor’s Theorem & Approximations
↭ L’Hôpital’s Rule for Indeterminate Forms
↭ Exam-Optimized Formula Sheets
↭ Fully Worked Past Paper Questions
↭ Common Mistakes & Examiner Traps
↭ One-Page Quick Reference
Based on LSE Course Materials 2024–25
Course Notes Chapter 1
Lecture Slides Week 2
Past Exam Papers 2023–2025
London School of Economics
Department of Mathematics
Academic Year 2025–26
,MA221 Week 2 Complete Guide 1
Contents
1 Week 2: Continuity, Taylor Expansion & L’Hôpital’s Rule 2
1.1 A. Core Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 What is Continuity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Why Does Continuity Matter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 What is Taylor’s Theorem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 What is L’Hôpital’s Rule? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 B. Rigorous Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 C. Key Theorems (Exam-Critical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 D. Formula & Identity Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Continuity Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Taylor’s Theorem Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.3 L’Hôpital’s Rule Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 E. Exam-Style Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 F. Fully Worked Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 G. Common Mistakes (Examiner Traps) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 H. One-Page Cheat Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Additional Practice Problems 21
2.1 Quick Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Solutions to Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
,MA221 Week 2 Complete Guide 2
1 Week 2: Continuity, Taylor Expansion & L’Hôpital’s Rule
1.1 A. Core Theory
1.1.1 What is Continuity?
Informally, a function is continuous if you can draw its graph without lifting your pen. More rigorously,
continuity means the limit equals the function value.
KEY INSIGHT
Continuity combines two concepts:
1. The limit lim f (x) exists
x→c
2. The function value f (c) exists
3. They are equal: lim f (x) = f (c)
x→c
No abrupt jumps, breaks, or holes!
1.1.2 Why Does Continuity Matter?
Continuous functions have special properties:
Intermediate Value Theorem (IVT): If f is continuous on [a, b], then f takes every value between
f (a) and f (b)
Extreme Value Theorem (EVT): Continuous functions on closed intervals attain maximum and
minimum values
We can swap limits and function evaluations
Integrals are well-defined (Week 3)
1.1.3 What is Taylor’s Theorem?
Taylor’s Theorem provides polynomial approximations to functions. Instead of working with compli-
cated functions like ex or sin(x), we can approximate them with polynomials!
Why polynomials?
Easy to evaluate (just arithmetic)
Easy to di!erentiate and integrate
Easy to analyze
Can approximate any smooth function arbitrarily well
INTUITION: Taylor Approximation
Think of Taylor polynomials as building increasingly accurate approximations:
0th degree: P0 (x) = f (c) (constant, just the function value)
1st degree: P1 (x) = f (c) + f ↑ (c)(x → c) (tangent line)
→→
2nd degree: P2 (x) = f (c) + f ↑ (c)(x → c) + f 2(c) (x → c)2 (parabola matching curvature)
Each degree adds more accuracy by matching another derivative!
, MA221 Week 2 Complete Guide 3
WHY TAYLOR WORKS: Approximating ex
The Taylor polynomial for ex about x = 0 is:
x2 x3 xn
Pn (x) = 1 + x + + + ··· +
2! 3! n!
Let’s see how accurate this is at x = 1 (approximating e ↑ 2.71828):
P0 (1) = 1 (o! by 63%)
P1 (1) = 2 (o! by 26%)
P2 (1) = 2.5 (o! by 8%)
P3 (1) = 2.667 (o! by 2%)
P5 (1) = 2.7167 (o! by 0.06%!)
Higher degree ↓ better approximation!
1.1.4 What is L’Hôpital’s Rule?
L’Hôpital’s Rule is a powerful tool for evaluating indeterminate forms like 00 or ↓
↓ . Instead of algebraic
manipulation, you di!erentiate the numerator and denominator separately!
Why it works: When both f (x) and g(x) approach 0, the ratio fg(x) (x)
behaves like the ratio of their
f → (x)
rates of change, which is g → (x) .
QUICK EXAMPLE: L’Hôpital in Action
sin(x)
Evaluate lim
x→0 x
Without L’Hôpital: Would need Taylor series or geometric arguments (messy!)
With L’Hôpital:
Check form: As x ↔ 0, both sin(x) ↔ 0 and x ↔ 0 ↓ Form is 0
0 ↭
sin(x) cos(x)
Apply rule: lim = lim =1
x→0 x x→0 1
Done! Much easier than other methods.
When NOT to use L’Hôpital:
When the form is NOT indeterminate (direct substitution works)
When algebraic simplification is easier (factoring, rationalizing)
When you’re asked to use a specific method (Taylor, definition, etc.)
1.2 B. Rigorous Definitions
Definition 2.1: Continuity at a Point
A function f is continuous at x = c if:
lim f (x) = f (c)
x→c
Equivalently, using the ω-ε definition:
