Calculus I Exam 1 Study Guide: Limits, Continuity, Asymptotes, and Derivatives

Limits and Their Properties

Average vs. Instantaneous Rate of Change

The average rate of change of a function $f(x)$ over an interval $[a, b]$ measures the change in $f(x)$ per unit change in $x$ across the interval. The instantaneous rate of change at a point is the derivative, representing how $f(x)$ changes at a specific value of $x$.

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

• Instantaneous Rate of Change: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Example: For $f(x) = x^2$, average rate from $x=1$ to $x=3$ is $\frac{9-1}{3-1} = 4$; instantaneous rate at $x=2$ is $f'(2) = 4$.

Calculating Limits: First Steps and Indeterminate Forms

When calculating limits, the first step is to substitute the value into the function. If this leads to an indeterminate form (such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$), further algebraic manipulation or limit laws are needed.

• Indeterminate Forms: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$

• Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ yields $\frac{0}{0}$; factor numerator to get $\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} x+2 = 4$.

Techniques for Evaluating Limits

• Direct Substitution: Substitute the value directly if the function is continuous at that point.

• Factoring: Factor numerator and denominator to cancel common terms.

• Rationalization: Multiply by the conjugate to simplify expressions with roots.

• Trigonometric Limits: Use known limits such as $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

• Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Asymptotes and Graphing

Vertical and Horizontal Asymptotes

An asymptote is a line that a graph approaches but never touches. Vertical asymptotes occur where the function grows without bound as $x$ approaches a certain value. Horizontal asymptotes describe the behavior of a function as $x$ approaches infinity.

• Vertical Asymptote: If $\lim_{x \to a} f(x) = \pm \infty$, then $x=a$ is a vertical asymptote.

• Horizontal Asymptote: If $\lim_{x \to \infty} f(x) = L$, then $y=L$ is a horizontal asymptote.

• Example: $f(x) = \frac{1}{x}$ has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

Sketching Graphs and Identifying Asymptotes

• Identify points where the denominator is zero for vertical asymptotes.

• Compare degrees of numerator and denominator for horizontal asymptotes.

• For rational functions $f(x) = \frac{P(x)}{Q(x)}$:

  • If degree of $P(x)$ < degree of $Q(x)$, $y=0$ is a horizontal asymptote.

  • If degrees are equal, $y=\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$.

  • If degree of $P(x)$ > degree of $Q(x)$, no horizontal asymptote.

Continuity and Types of Discontinuity

Definition of Continuity

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

• Removable Discontinuity: Occurs when $\lim_{x \to a} f(x)$ exists but $f(a)$ is not defined or not equal to the limit.

• Jump Discontinuity: The left and right limits exist but are not equal.

• Infinite Discontinuity: The function approaches infinity at $x=a$.

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x=1$ has a removable discontinuity.

The Squeeze Theorem

Statement and Application

The Squeeze Theorem is used to find limits of functions trapped between two other functions whose limits are known.

• If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $a$, and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

• Example: $\lim_{x \to 0} x^2 \sin(1/x) = 0$

The Intermediate Value Theorem (IVT)

Statement and Use

The Intermediate Value Theorem states that if $f(x)$ is continuous on $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists $c$ in $[a, b]$ such that $f(c) = N$.

• Used to show the existence of roots or solutions within an interval.

• Example: If $f(1) = -2$ and $f(3) = 4$, then $f(x)$ must be zero for some $x$ in $[1, 3]$.

Introducing the Derivative

Definition and Geometric Interpretation

The derivative of $f(x)$ at $x=a$ is the instantaneous rate of change of $f(x)$ at $a$, and is the slope of the tangent line to the graph at that point.

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

• Geometric Meaning: Slope of the tangent line to $f(x)$ at $x=a$

• Example: For $f(x) = x^2$, $f'(a) = 2a$

Equation of a Tangent Line

• The equation for the tangent line at $x=a$ is $y = f(a) + f'(a)(x - a)$

• Example: For $f(x) = x^2$ at $x=1$, $f(1) = 1$, $f'(1) = 2$, so $y = 1 + 2(x - 1)$

Derivative and Tangent Line Slope

• The derivative at a point gives the slope of the tangent line at that point.

• If $f'(a)$ does not exist, the graph is not differentiable at $x=a$.

• Example: $f(x) = |x|$ is not differentiable at $x=0$.

Factorization and Graphing Techniques

Factoring Polynomials

• Factor expressions to simplify limits and find roots.

• Example: $x^2 - 9 = (x - 3)(x + 3)$

Graphing Non-Differentiable Functions

• Points where the derivative does not exist may be cusps, corners, or vertical tangents.

• Example: $f(x) = |x|$ has a corner at $x=0$.

Summary Table: Types of Discontinuity

Type	Description	Example
Removable	Limit exists, function not defined or not equal to limit	$f(x) = \frac{x^2 - 1}{x - 1}$ at $x=1$
Jump	Left and right limits exist but are not equal	Piecewise function with different values at $x=a$
Infinite	Function approaches infinity at $x=a$	$f(x) = \frac{1}{x}$ at $x=0$

Practice and Preparation

• Review exercises from textbook sections 2.1–2.6 and 3.1–3.5.

• Focus on understanding concepts, not just memorizing procedures.

• Practice sketching graphs, identifying asymptotes, and calculating limits and derivatives.

Additional info: Some context and examples have been inferred and expanded for clarity and completeness based on standard Calculus I curriculum.