Limits and Continuity: Study Guide for Calculus I

Limits and Continuity in Calculus

Introduction

This study guide covers foundational concepts in Calculus I, focusing on limits, continuity, and the use of graphical and analytical methods to evaluate limits and determine continuity. These topics are essential for understanding the behavior of functions and for the development of differential calculus.

Limits

Definition of a Limit

The limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets closer to a. It is denoted as:

• $\lim_{x \to a} f(x)$

If f(x) approaches the same value from both the left and right as x approaches a, the limit exists.

Evaluating Limits Numerically

• Limits can be estimated by creating a table of values for f(x) as x approaches a from both sides.

• Example: To estimate $\lim_{x \to 0} \frac{\sqrt{x+3} - \sqrt{3}}{x}$, substitute values close to 0 (e.g., -0.1, -0.01, 0.01, 0.1) and observe the trend in f(x).

Evaluating Limits Analytically

• Direct Substitution: Substitute a into f(x). If the result is defined, that is the limit.

• Factoring: If direct substitution yields an indeterminate form (like 0/0), factor and simplify the expression before substituting.

• Rationalization: Multiply numerator and denominator by a conjugate if the limit involves square roots.

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

One-Sided Limits

• The left-hand limit is $\lim_{x \to a^-} f(x)$.

• The right-hand limit is $\lim_{x \to a^+} f(x)$.

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

Limits Involving Infinity

• Limits can approach infinity if the function increases or decreases without bound as x approaches a value.

• Example: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$

Piecewise Functions and Limits

• For piecewise functions, evaluate the limit from each side using the appropriate piece.

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

Examples

• $\lim_{x \to 2} (5 - 2x) = 5 - 2(2) = 1$

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

• $\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$: Factor numerator to get $\frac{(x-4)(x+4)}{x-4} = x+4$ for $x \neq 4$, so the limit is $4+4=8$.

Continuity

Definition of Continuity at a Point

A function f(x) is continuous at x = a if:

1. f(a) is defined.

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

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

Types of Discontinuities

• Removable Discontinuity: The limit exists, but f(a) is not defined or $f(a) \neq \lim_{x \to a} f(x)$.

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

• Infinite Discontinuity: The function approaches infinity at a.

Determining Continuity from a Graph

• Look for holes (removable), jumps, or vertical asymptotes (infinite) in the graph.

• Check if the function is defined at the point and if the limit matches the function value.

Intervals of Continuity

• A function is continuous on an interval if it is continuous at every point in that interval.

• For rational functions, discontinuities occur where the denominator is zero.

Graphical Analysis of Limits and Continuity

Using Tables and Graphs

• Tables help estimate limits numerically.

• Graphs help visualize where limits exist, where discontinuities occur, and the type of discontinuity.

Example Table (Numerical Estimation of a Limit)

x	-0.1	-0.01	-0.001	0.001	0.01	0.1
f(x)	...	...	...	...	...	...

Additional info: Fill in the table by substituting each x-value into the function and rounding to four decimal places as required.

Limit Definition of the Derivative

Definition

The derivative of f(x) at x = a is defined as:

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

This gives the slope of the tangent line to the graph of f(x) at x = a.

Example

• Given $f(x) = x^2$, $f'(a) = \lim_{h \to 0} \frac{(a+h)^2 - a^2}{h} = \lim_{h \to 0} \frac{2ah + h^2}{h} = \lim_{h \to 0} (2a + h) = 2a$

Asymptotes

Vertical and Horizontal Asymptotes

• Vertical asymptotes occur where the function approaches infinity as x approaches a certain value (usually where the denominator is zero).

• Horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity.

Example Table: Asymptotes of Rational Functions

Function	Vertical Asymptote(s)	Horizontal Asymptote(s)
$\frac{1}{x-2}$	$x=2$	$y=0$
$\frac{x-2}{x^2+x-6}$	$x=2, x=-3$	$y=0$

Summary Table: Types of Discontinuities

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

Practice Problems and Applications

• Estimate limits using tables and graphs.

• Find limits analytically using algebraic techniques.

• Identify points of discontinuity and classify them as removable or non-removable.

• Use the limit definition of the derivative to find the slope of the tangent line.

• Determine equations of tangent lines using the point-slope form.

Key Formulas

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

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

• Point-slope form of a line: $y - y_1 = m(x - x_1)$

Additional info:

• Some problems require sketching graphs based on given limit and continuity properties.

• Students are expected to distinguish between removable and non-removable discontinuities and to use interval notation for domains of continuity.

• Tables and graphs are used extensively for both estimation and conceptual understanding.