Calculus I: Limits and Continuity – Problem Set Study Guide

Limits and Continuity

Introduction

This study guide covers fundamental concepts in limits and continuity, as presented in a typical Calculus I problem set. Understanding limits is essential for analyzing the behavior of functions near specific points and forms the foundation for the study of derivatives and integrals.

Evaluating Limits

Definition of a Limit

• Limit: The value that a function f(x) approaches as x approaches a particular point a.

• Notation: $\lim_{x \to a} f(x) = L$ means that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L.

Evaluating Limits Algebraically

• Direct Substitution: If f(x) is continuous at a, then $\lim_{x \to a} f(x) = f(a)$.

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

• Rationalization: Multiply by a conjugate to simplify expressions involving roots.

• Special Limits: Recognize standard limits such as $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

One-Sided Limits

• Left-hand limit: $\lim_{x \to a^-} f(x)$ is the value as x approaches a from the left.

• Right-hand limit: $\lim_{x \to a^+} f(x)$ is the value as x approaches a from the right.

• The two-sided limit exists only if both one-sided limits exist and are equal.

Limits Involving Piecewise Functions

• For piecewise functions, evaluate the limit from each side using the appropriate formula for f(x).

• If the left- and right-hand limits are not equal, the limit does not exist at that point.

Examples

• Example 1: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ Factor numerator: $x^2 - 4 = (x-2)(x+2)$ Simplify: $\frac{(x-2)(x+2)}{x-2} = x+2$ (for $x \neq 2$) Substitute: $x = 2 \implies 2+2 = 4$ Answer: 4

• Example 2: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (standard limit)

Determining When Limits Do Not Exist

• Limits may not exist if:

  • The left- and right-hand limits are not equal.

  • The function grows without bound (infinite limit).

  • The function oscillates as x approaches a.

Limits Involving Parameters

Finding Values for Existence of Limits

• Sometimes, a limit will exist only for certain values of a parameter (e.g., $k$ in $\lim_{x \to 3} \frac{5x - 6k + 1}{x - 3}$).

• Set the numerator to zero at the problematic point to ensure the limit exists and solve for the parameter.

Continuity and Removable Discontinuities

Definition of Continuity

• Continuous at a point: f(x) is continuous at x = a if:

  • $f(a)$ is defined

  • $\lim_{x \to a} f(x)$ exists

  • $\lim_{x \to a} f(x) = f(a)$

• Removable discontinuity: Occurs when the limit exists but $f(a)$ is not defined or not equal to the limit.

Special Trigonometric and Radical Limits

• Use trigonometric identities and standard limits for expressions involving $\sin x$, $\cos x$, etc.

• For radicals, rationalize the numerator or denominator as needed.

Limits at Infinity and Infinite Limits

• As x approaches infinity, compare the degrees of numerator and denominator in rational functions:

  • If degrees are equal: limit is the ratio of leading coefficients.

  • If numerator degree < denominator degree: limit is 0.

  • If numerator degree > denominator degree: limit does not exist (infinite).

Average and Instantaneous Velocity

Definitions

• Average velocity: $\frac{s(t_2) - s(t_1)}{t_2 - t_1}$, where $s(t)$ is the position function.

• Instantaneous velocity: The derivative $s'(t)$, or $\lim_{h \to 0} \frac{s(t+h) - s(t)}{h}$.

Example

• Given $s(t) = \frac{1}{2}t^2 - 2\sqrt{t}$, find average velocity from $t=0$ to $t=4$:

• Compute $s(4)$ and $s(0)$, then use the average velocity formula.

• For instantaneous velocity at $t=4$, compute $s'(t)$ and evaluate at $t=4$.

Sample Table: Types of Discontinuities

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

Summary of Key Formulas

• Limit definition: $\lim_{x \to a} f(x)$

• Average velocity: $\frac{s(t_2) - s(t_1)}{t_2 - t_1}$

• Instantaneous velocity: $\lim_{h \to 0} \frac{s(t+h) - s(t)}{h}$

• Standard trigonometric limits:

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

  • $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

Practice Problems

• Evaluate limits using algebraic simplification, factoring, and rationalization.

• Determine values of parameters for which limits exist.

• Analyze piecewise functions for continuity and removable discontinuities.

• Apply limit definitions to velocity problems.