Calculus is the mathematics of change. Limits, continuity, and differentiability form the foundation of differential calculus.
Overview
graph TD
A[Calculus Fundamentals] --> B[Limits]
A --> C[Continuity]
A --> D[Differentiability]
B --> B1[Standard Forms]
B --> B2[L'Hopital's Rule]
C --> C1[Types of Discontinuity]
D --> D1[Derivatives]
D --> D2[Applications]Limits
Interactive Demo: Function Plotter
Explore different mathematical functions and their graphs:
Definition
$$\lim_{x \to a} f(x) = L$$means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.
Left and Right Hand Limits
- LHL: $\lim_{x \to a^-} f(x)$ (approaching from left)
- RHL: $\lim_{x \to a^+} f(x)$ (approaching from right)
Limit exists if and only if: $\text{LHL} = \text{RHL}$
Properties of Limits
If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:
| Property | Formula |
|---|---|
| Sum | $\lim[f(x) + g(x)] = L + M$ |
| Product | $\lim[f(x) \cdot g(x)] = L \cdot M$ |
| Quotient | $\lim\frac{f(x)}{g(x)} = \frac{L}{M}$ (M ≠ 0) |
| Power | $\lim[f(x)]^n = L^n$ |
Standard Limits
Algebraic Limits
$$\boxed{\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}}$$Trigonometric Limits
$$\boxed{\lim_{x \to 0} \frac{\sin x}{x} = 1}$$ $$\boxed{\lim_{x \to 0} \frac{\tan x}{x} = 1}$$ $$\boxed{\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}}$$Exponential and Logarithmic Limits
$$\boxed{\lim_{x \to 0} \frac{e^x - 1}{x} = 1}$$ $$\boxed{\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a}$$ $$\boxed{\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1}$$ $$\boxed{\lim_{x \to 0} (1 + x)^{1/x} = e}$$ $$\boxed{\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e}$$Indeterminate Forms
| Form | Type |
|---|---|
| $\frac{0}{0}$ | Indeterminate |
| $\frac{\infty}{\infty}$ | Indeterminate |
| $0 \times \infty$ | Indeterminate |
| $\infty - \infty$ | Indeterminate |
| $0^0$ | Indeterminate |
| $1^\infty$ | Indeterminate |
| $\infty^0$ | Indeterminate |
L’Hôpital’s Rule
For $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$(if the right side exists)
Continuity
Definition
A function $f(x)$ is continuous at $x = a$ if:
- $f(a)$ is defined
- $\lim_{x \to a} f(x)$ exists
- $\lim_{x \to a} f(x) = f(a)$
Types of Discontinuity
graph TD
A[Discontinuity] --> B[Removable]
A --> C[Non-removable]
B --> B1[Limit exists but ≠ f(a)]
C --> C1[Jump: LHL ≠ RHL]
C --> C2[Infinite: limit is ∞]
C --> C3[Oscillatory: no limit]Properties of Continuous Functions
- Sum, difference, product of continuous functions is continuous
- Quotient is continuous where denominator ≠ 0
- Composition of continuous functions is continuous
Intermediate Value Theorem
If $f$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = k$.
Differentiability
Definition
$f(x)$ is differentiable at $x = a$ if:
$$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$exists.
Relation: Differentiability ⟹ Continuity
If $f$ is differentiable at $a$, then $f$ is continuous at $a$.
But the converse is NOT true!
Example: $f(x) = |x|$ is continuous at $x = 0$ but not differentiable.
Differentiation Rules
Basic Rules
| Function | Derivative |
|---|---|
| $c$ (constant) | $0$ |
| $x^n$ | $nx^{n-1}$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\log_a x$ | $\frac{1}{x \ln a}$ |
Trigonometric Derivatives
| Function | Derivative |
|---|---|
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
Inverse Trigonometric Derivatives
| Function | Derivative |
|---|---|
| $\sin^{-1} x$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\cos^{-1} x$ | $\frac{-1}{\sqrt{1-x^2}}$ |
| $\tan^{-1} x$ | $\frac{1}{1+x^2}$ |
| $\cot^{-1} x$ | $\frac{-1}{1+x^2}$ |
Rules of Differentiation
Product Rule
$$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)$$Quotient Rule
$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$Chain Rule
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$Applications of Derivatives
Rate of Change
If $y = f(x)$, then $\frac{dy}{dx}$ represents the rate of change of $y$ with respect to $x$.
Monotonicity
- $f'(x) > 0$ in $(a,b)$ ⟹ $f$ is increasing in $(a,b)$
- $f'(x) < 0$ in $(a,b)$ ⟹ $f$ is decreasing in $(a,b)$
Maxima and Minima
First Derivative Test:
- $f'(x)$ changes from + to - at $x = c$ ⟹ Local maximum
- $f'(x)$ changes from - to + at $x = c$ ⟹ Local minimum
Second Derivative Test: At critical point $c$ where $f'(c) = 0$:
- $f''(c) < 0$ ⟹ Local maximum
- $f''(c) > 0$ ⟹ Local minimum
Practice Problems
Evaluate: $\lim_{x \to 0} \frac{\sin 3x}{\tan 2x}$
Find: $\lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{1/x^2}$
Check continuity and differentiability of $f(x) = |x - 1| + |x - 2|$ at $x = 1, 2$.
Find maxima and minima of $f(x) = x^3 - 3x^2 - 9x + 5$.