Limits, Derivatives, Integrals, Applications & Practice Problems
1. Introduction
• Calculus studies change (derivatives) and accumulation (integrals).
• Branches:
1. Differential Calculus – Rates of change, slope of curves.
2. Integral Calculus – Area under curves, accumulation.
• Applications: Physics, Economics, Engineering, Biology, Computer Science.
2. Limits and Continuity
• :
3. Differentiation
• Definition:
• Rules:
o Power Rule:
o Constant Rule:
Absolutely! Here's a complete, well-structured set of notes on Limits and Continuity with
examples and graphs. I’ll organize it in a way that’s easy to study and includes visual
explanations.
Limits and Continuity – Complete Notes
1. Introduction
Calculus is the study of change and accumulation.
Two fundamental concepts in calculus are:
1. Limits – Understanding the behavior of a function as it approaches a point.
2. Continuity – Understanding if a function has no “breaks” at a point.
, 2. Limits
2.1 Definition
The limit of a function as approaches is the value that gets closer to, as gets closer to .
\lim_{x \to a} f(x) = L
• is the limit.
• This does not require to exist.
Example 1:
f(x) = 2x + 3
\lim_{x \to 2} (2x + 3) = 2(2) + 3 = 7
2.2 One-Sided Limits
• Left-hand limit (LHL):
• Right-hand limit (RHL):
A limit exists only if:
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)
Example 2:
f(x) =
\begin{cases}
x+2 & x < 1 \\
3x & x \ge 1
\end{cases}
\lim_{x \to 1^-} f(x) = 1+2 = 3
\lim_{x \to 1^+} f(x) = 3(1) = 3
Graph:
Step function approaching same value from left and right.
1. Introduction
• Calculus studies change (derivatives) and accumulation (integrals).
• Branches:
1. Differential Calculus – Rates of change, slope of curves.
2. Integral Calculus – Area under curves, accumulation.
• Applications: Physics, Economics, Engineering, Biology, Computer Science.
2. Limits and Continuity
• :
3. Differentiation
• Definition:
• Rules:
o Power Rule:
o Constant Rule:
Absolutely! Here's a complete, well-structured set of notes on Limits and Continuity with
examples and graphs. I’ll organize it in a way that’s easy to study and includes visual
explanations.
Limits and Continuity – Complete Notes
1. Introduction
Calculus is the study of change and accumulation.
Two fundamental concepts in calculus are:
1. Limits – Understanding the behavior of a function as it approaches a point.
2. Continuity – Understanding if a function has no “breaks” at a point.
, 2. Limits
2.1 Definition
The limit of a function as approaches is the value that gets closer to, as gets closer to .
\lim_{x \to a} f(x) = L
• is the limit.
• This does not require to exist.
Example 1:
f(x) = 2x + 3
\lim_{x \to 2} (2x + 3) = 2(2) + 3 = 7
2.2 One-Sided Limits
• Left-hand limit (LHL):
• Right-hand limit (RHL):
A limit exists only if:
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)
Example 2:
f(x) =
\begin{cases}
x+2 & x < 1 \\
3x & x \ge 1
\end{cases}
\lim_{x \to 1^-} f(x) = 1+2 = 3
\lim_{x \to 1^+} f(x) = 3(1) = 3
Graph:
Step function approaching same value from left and right.
