Limits, Continuity, and Differentiation: Calculus I Study Guide

Limits, Continuity, and Differentiation

Introduction

This study guide covers foundational concepts in Calculus I, including limits, continuity, and differentiation. It provides definitions, properties, and examples, as well as practice problems and their solutions. These topics are essential for understanding the behavior of functions and the principles of calculus.

Limits

Definition of a Limit

• Informal Definition: If f(x) becomes arbitrarily close to a single number L as x approaches c from both sides, then the limit of f(x) as x approaches c is L.

This is written as:

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

• Formal (Epsilon-Delta) Definition: Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. We say that $\lim_{x \to c} f(x) = L$ if for every number $\varepsilon > 0$ there exists a corresponding number $\delta > 0$ such that $|f(x) - L| < \varepsilon$ whenever $0 < |x - c| < \delta$.

Evaluating Limits Analytically

• Direct substitution is often used when the function is continuous at the point.

• If direct substitution yields an indeterminate form (such as $0/0$), algebraic manipulation (factoring, rationalizing, etc.) may be necessary.

Evaluating Limits Numerically

• Limits can be estimated using tables of values approaching the point of interest from both sides.

Examples

• $\lim_{x \to 3} x^2 = 9$

• $\lim_{x \to 2} \frac{x^3 - 7x + 12}{x-2}$ (requires factoring and simplification)

• $\lim_{x \to 4} \frac{\sqrt{x+1} - 3}{x-4}$ (requires rationalizing the numerator)

Continuity

Definition of Continuity at a Point

A function f is continuous at a point $x = c$ if and only if all three of the following conditions are met:

1. $f(c)$ exists (c is in the domain of f)

2. $\lim_{x \to c} f(x)$ exists (f has a limit as x approaches c)

3. $\lim_{x \to c} f(x) = f(c)$ (the limit equals the function value)

Types of Discontinuities

• Removable Discontinuity: The limit exists, but the function is not defined at that point or $f(c) \neq \lim_{x \to c} f(x)$.

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

• Infinite Discontinuity: The function approaches infinity at the point.

Example Table: Conditions for Continuity

Condition	Description
1. $f(c)$ exists	c is in the domain of f
2. $\lim_{x \to c} f(x)$ exists	f has a limit as x approaches c
3. $\lim_{x \to c} f(x) = f(c)$	The limit equals the function value

Differentiation

Definition of the Derivative

The derivative of the function f(x) with respect to the variable x is the function f'(x) whose value at x is:

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

• If this limit exists, f is said to be differentiable at x.

When is a Function Not Differentiable at a Point?

• f is not continuous at c

• f has a corner or cusp at c

• f has a vertical tangent at c

• f has infinite oscillations approaching c

Example: Using the Definition of the Derivative

• Find $\frac{d}{dx}(4x^2 - 4x + 1)$ using the definition:

$f'(x) = \lim_{h \to 0} \frac{4(x+h)^2 - 4(x+h) + 1 - [4x^2 - 4x + 1]}{h}$

Simplify and evaluate the limit to find the derivative.

Graphical Analysis

Continuity and Discontinuity on Graphs

• Identify points where the function is not continuous.

• At each discontinuity, state which condition of continuity is violated (see table above).

Limits from Graphs

• Estimate $\lim_{x \to a^-} f(x)$, $\lim_{x \to a^+} f(x)$, and $\lim_{x \to a} f(x)$ from the graph.

Practice Problems and Solutions

Sample Limit Problems

• $\lim_{x \to 3} x^2 = 9$

• $\lim_{x \to 2} \frac{x^3 - 7x + 12}{x-2} = 1$

• $\lim_{x \to 4} \frac{\sqrt{x+1} - 3}{x-4} = \frac{1}{6}$

• $\lim_{x \to 1} \frac{1}{x-1}$ does not exist (infinite discontinuity)

Sample Derivative Problems

• Find $\frac{dy}{dx}$ for $y = 4x^2 - 4x + 1$ using the definition of the derivative.

• Find $\frac{dy}{dx}$ for $y = x^2 \sin x + \cos x$ using product and chain rules.

Related Rates and Motion Problems

• Related Rates: A spherical balloon is inflated at a rate of 100 cubic feet per minute. How fast is the radius increasing when the radius is 4 feet?

• Motion: The position of a particle is given by $x(t) = 4 \sin(2t) + 2 \cos(3t)$. Find the acceleration at $t = \frac{\pi}{5}$.

Summary Table: Types of Discontinuities

Type	Description	Graphical Feature
Removable	Limit exists, function value missing or not equal to limit	Hole in the graph
Jump	Left and right limits exist but are not equal	Step/jump in the graph
Infinite	Function approaches infinity	Vertical asymptote

Key Formulas

• Limit Definition: $\lim_{x \to c} f(x) = L$

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

• Continuity at a Point: $\lim_{x \to c} f(x) = f(c)$

Additional info:

• Some problems require the use of tables to estimate limits numerically.

• Graphical analysis is important for understanding continuity and differentiability.

• Practice problems include both computational and conceptual questions, as well as applications to motion and related rates.