Calculus I: Limits, Continuity, and Derivatives – Study Notes

Chapter 2: Limits & Continuity

Section 2.1 – Rates of Change and Tangent Lines to Curves

Understanding the concept of a limit is fundamental to calculus, especially for defining derivatives and integrals. This section introduces the average and instantaneous rates of change, leading to the definition of the derivative.

• Average Rate of Change (Slope): For a function f(x) between two points (x₁, y₁) and (x₂, y₂):

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Instantaneous Rate of Change: As the interval shrinks ($x_2 \to x_1$), the average rate approaches the instantaneous rate, which is the derivative.

• Secant vs. Tangent: The secant line connects two points; the tangent line touches the curve at one point and represents the instantaneous rate of change.

Example: Find the slope of $f(x) = x^2 - 4x$ at $P(1, -3)$ using the average rate of change definition.

Section 2.2 – Limits of a Function & Limit Laws

Limits describe the behavior of a function as the input approaches a certain value. They are essential for defining derivatives and continuity.

• Definition of a Limit: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

• Situations Where Limits Do Not Exist:

  • Jumps

  • Grows too large (infinite limits)

  • Oscillates too much

Basic Limit Laws:

Law	Formula
Sum/Difference	$\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$
Constant Multiple	$\lim_{x \to a} [k f(x)] = k \lim_{x \to a} f(x)$
Product	$\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
Quotient	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, $\lim_{x \to a} g(x) \neq 0$
Power/Root	$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$

Limits of Polynomials and Rational Functions:

• For polynomials: $\lim_{x \to a} P(x) = P(a)$

• For rational functions: $\lim_{x \to a} \frac{P(x)}{Q(x)} = \frac{P(a)}{Q(a)}$ if $Q(a) \neq 0$

Manipulating Limits: When direct substitution gives $0/0$, factor or use conjugates to simplify.

Section 2.3 – The Precise Definition of a Limit

The formal (epsilon-delta) definition of a limit provides mathematical rigor to the concept of limits.

• For every $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - L| < \varepsilon$ whenever $0 < |x - c| < \delta$.

Example: Show $\lim_{x \to 2} (5x - 3) = 7$ using the precise definition.

Section 2.4 – One-Sided Limits

One-sided limits consider the behavior of a function as $x$ approaches a value from only one side (left or right).

• Left-hand limit: $\lim_{x \to c^-} f(x)$

• Right-hand limit: $\lim_{x \to c^+} f(x)$

• A limit exists at a point if and only if both one-sided limits exist and are equal.

Section 2.5 – Continuity

A function is continuous at $x = c$ if:

1. $f(c)$ is defined

2. $\lim_{x \to c} f(x)$ exists

3. $\lim_{x \to c} f(x) = f(c)$

Types of Discontinuity:

• Removable

• Jump

• Infinite

• Oscillating

Properties of Continuous Functions: Sums, differences, products, quotients (where denominator is nonzero), and compositions of continuous functions are continuous.

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $[a, b]$ such that $f(c) = N$.

Section 2.8 – Limits Involving Infinity & Asymptotes of Graphs

As $x$ approaches infinity, the behavior of rational functions can be analyzed to find horizontal and oblique asymptotes.

• Divide numerator and denominator by the highest power of $x$ in the denominator.

• Vertical asymptotes occur where the denominator is zero (and not canceled by the numerator).

• Horizontal asymptotes are determined by the degrees of numerator and denominator.

Chapter 3: Derivatives

Section 3.1 – Tangent Line and the Derivative at a Point

The derivative at a point gives the slope of the tangent line to the curve at that point, representing the instantaneous rate of change.

• Definition: $f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

• This is also called the limit of the difference quotient.

Example: For $f(x) = \frac{1}{x}$, find the slope at $x = -1$ and the equation of the tangent line.

Section 3.2 – The Derivative as a Function

The derivative function $f'(x)$ gives the slope of the tangent line at any point $x$ on the curve.

• Definition: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Alternative form: $f'(x) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Derivative Notations:

• $f'(x)$

• $\frac{dy}{dx}$

• $\frac{d}{dx} f(x)$

• $D_x f(x)$

Example: Calculate the derivative of $f(x) = 3x - 5$ using the limit definition.

When Does a Function Not Have a Derivative at a Point?

• Sharp corners (cusps)

• Vertical tangent lines

• Discontinuities

• Wild oscillations

If a function is differentiable at $x = c$, then it is continuous at $x = c$. The converse is not always true.

Relating a Function and Its Derivative

The graph of the derivative provides information about the slope of the original function at each point.

Summary Table: Types of Discontinuity

Type	Description
Removable	Hole in the graph; limit exists but function value is different or undefined
Jump	Function "jumps" to a new value; left and right limits are different
Infinite	Function approaches infinity at a point
Oscillating	Function oscillates infinitely near a point

Key Formulas

• Average Rate of Change: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Derivative at a Point: $f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$

• Derivative Function: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Limit Laws: See table above

Additional info: These notes cover the foundational concepts of limits, continuity, and derivatives, including formal definitions, graphical interpretations, and key theorems such as the Intermediate Value Theorem and the Sandwich Theorem. Examples and exercises are provided throughout to reinforce understanding.