Calculus: A Comprehensive Exploration
Introduction
Welcome to the fascinating world of calculus! This branch of mathematics provides the tools
to understand and model change. It has wide-ranging applications in various fields like
physics, engineering, economics, and computer science. This course will cover the
fundamental concepts of calculus, including limits, derivatives, and integrals.
Chapter 1: Limits and Continuity
1.1 Limits
● Definition: The limit of a function f(x) as x approaches a value c is the value that f(x)
gets closer and closer to as x gets closer and closer to c.
● Notation: lim (x→c) f(x) = L
● Epsilon-Delta Definition (More Rigorous):
For every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε
● Techniques for finding limits:
○ Direct substitution
○ Factoring
○ Rationalizing
○ L'Hôpital's Rule (for indeterminate forms)
○ Squeeze Theorem
● One-Sided Limits:
○ Limit from the left: lim (x→c-) f(x)
○ Limit from the right: lim (x→c+) f(x)
● Infinite Limits and Limits at Infinity:
○ Vertical asymptotes
○ Horizontal asymptotes
Example 1: Find the limit of the function f(x) = (x² - 1) / (x - 1) as x approaches 1.
Solution:
lim (x→1) (x² - 1) / (x - 1) = lim (x→1) (x + 1)(x - 1) / (x - 1) = lim (x→1) (x + 1) = 1 + 1 = 2
Example 2: Find the limit of the function f(x) = sin(x)/x as x approaches 0.
Solution:
Using L'Hôpital's Rule:
lim (x→0) sin(x)/x = lim (x→0) cos(x)/1 = cos(0)/1 = 1
Example 3: Find the limit of the function f(x) = (√(x+1) - 1)/x as x approaches 0.
, Solution:
Rationalizing the numerator:
lim (x→0) (√(x+1) - 1)/x * (√(x+1) + 1)/(√(x+1) + 1) = lim (x→0) (x+1 - 1)/(x(√(x+1) + 1))
= lim (x→0) x/(x(√(x+1) + 1)) = lim (x→0) 1/(√(x+1) + 1) = 1/(√(0+1) + 1) = 1/2
Example 4: Find the limit of the function f(x) = x^2 * sin(1/x) as x approaches 0.
Solution:
Using the Squeeze Theorem:
Since -1 ≤ sin(1/x) ≤ 1, we have -x^2 ≤ x^2 * sin(1/x) ≤ x^2
lim (x→0) -x^2 = 0 and lim (x→0) x^2 = 0
Therefore, by the Squeeze Theorem, lim (x→0) x^2 * sin(1/x) = 0
1.2 Continuity
● Definition: A function f(x) is continuous at x = c if:
1. f(c) is defined
2. lim (x→c) f(x) exists
3. lim (x→c) f(x) = f(c)
● Types of discontinuity:
○ Removable discontinuity (can be "removed" by redefining the function at that
point)
○ Jump discontinuity (the function "jumps" from one value to another)
○ Infinite discontinuity (the function approaches infinity)
○ Essential Discontinuity
● Continuity on an Interval:
A function is continuous on an open interval (a,b) if it is continuous at each point in the
interval. A function is continuous on a closed interval [a,b] if it is continuous on (a,b)
and continuous from the right at a and continuous from the left at b.
● The Intermediate Value Theorem:
If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b),
then there exists a number c in [a, b] such that f(c) = k.
Example 2: Determine the continuity of the function f(x) = { x², x ≤ 1; 2x - 1, x > 1 } at x = 1.
Solution:
1. f(1) = 1² = 1
2. lim (x→1-) f(x) = lim (x→1-) x² = 1
lim (x→1+) f(x) = lim (x→1+) 2x - 1 = 1
lim (x→1) f(x) = 1
3. lim (x→1) f(x) = f(1) = 1
Therefore, the function is continuous at x = 1.
Example 3: Determine the continuity of the function f(x) = 1/x at x = 0.
Solution:
f(0) is not defined.
Introduction
Welcome to the fascinating world of calculus! This branch of mathematics provides the tools
to understand and model change. It has wide-ranging applications in various fields like
physics, engineering, economics, and computer science. This course will cover the
fundamental concepts of calculus, including limits, derivatives, and integrals.
Chapter 1: Limits and Continuity
1.1 Limits
● Definition: The limit of a function f(x) as x approaches a value c is the value that f(x)
gets closer and closer to as x gets closer and closer to c.
● Notation: lim (x→c) f(x) = L
● Epsilon-Delta Definition (More Rigorous):
For every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε
● Techniques for finding limits:
○ Direct substitution
○ Factoring
○ Rationalizing
○ L'Hôpital's Rule (for indeterminate forms)
○ Squeeze Theorem
● One-Sided Limits:
○ Limit from the left: lim (x→c-) f(x)
○ Limit from the right: lim (x→c+) f(x)
● Infinite Limits and Limits at Infinity:
○ Vertical asymptotes
○ Horizontal asymptotes
Example 1: Find the limit of the function f(x) = (x² - 1) / (x - 1) as x approaches 1.
Solution:
lim (x→1) (x² - 1) / (x - 1) = lim (x→1) (x + 1)(x - 1) / (x - 1) = lim (x→1) (x + 1) = 1 + 1 = 2
Example 2: Find the limit of the function f(x) = sin(x)/x as x approaches 0.
Solution:
Using L'Hôpital's Rule:
lim (x→0) sin(x)/x = lim (x→0) cos(x)/1 = cos(0)/1 = 1
Example 3: Find the limit of the function f(x) = (√(x+1) - 1)/x as x approaches 0.
, Solution:
Rationalizing the numerator:
lim (x→0) (√(x+1) - 1)/x * (√(x+1) + 1)/(√(x+1) + 1) = lim (x→0) (x+1 - 1)/(x(√(x+1) + 1))
= lim (x→0) x/(x(√(x+1) + 1)) = lim (x→0) 1/(√(x+1) + 1) = 1/(√(0+1) + 1) = 1/2
Example 4: Find the limit of the function f(x) = x^2 * sin(1/x) as x approaches 0.
Solution:
Using the Squeeze Theorem:
Since -1 ≤ sin(1/x) ≤ 1, we have -x^2 ≤ x^2 * sin(1/x) ≤ x^2
lim (x→0) -x^2 = 0 and lim (x→0) x^2 = 0
Therefore, by the Squeeze Theorem, lim (x→0) x^2 * sin(1/x) = 0
1.2 Continuity
● Definition: A function f(x) is continuous at x = c if:
1. f(c) is defined
2. lim (x→c) f(x) exists
3. lim (x→c) f(x) = f(c)
● Types of discontinuity:
○ Removable discontinuity (can be "removed" by redefining the function at that
point)
○ Jump discontinuity (the function "jumps" from one value to another)
○ Infinite discontinuity (the function approaches infinity)
○ Essential Discontinuity
● Continuity on an Interval:
A function is continuous on an open interval (a,b) if it is continuous at each point in the
interval. A function is continuous on a closed interval [a,b] if it is continuous on (a,b)
and continuous from the right at a and continuous from the left at b.
● The Intermediate Value Theorem:
If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b),
then there exists a number c in [a, b] such that f(c) = k.
Example 2: Determine the continuity of the function f(x) = { x², x ≤ 1; 2x - 1, x > 1 } at x = 1.
Solution:
1. f(1) = 1² = 1
2. lim (x→1-) f(x) = lim (x→1-) x² = 1
lim (x→1+) f(x) = lim (x→1+) 2x - 1 = 1
lim (x→1) f(x) = 1
3. lim (x→1) f(x) = f(1) = 1
Therefore, the function is continuous at x = 1.
Example 3: Determine the continuity of the function f(x) = 1/x at x = 0.
Solution:
f(0) is not defined.
