Limits and Continuity: Key Concepts and Theorems in Calculus

Limits and Continuity

Limit of a Function and Limit Laws

The concept of a limit is foundational in calculus, describing the behavior of a function as its input approaches a particular value. Limit laws provide rules for calculating limits efficiently.

• Definition: 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.

• Limit Laws: These include the sum, difference, product, quotient, and power laws for limits.

• Example: $\lim_{x \to 2} (3x + 1) = 7$

One-Sided Limits

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

• Left-Hand Limit: $\lim_{x \to a^-} f(x)$

• Right-Hand Limit: $\lim_{x \to a^+} f(x)$

• Application: Useful for functions with jump discontinuities or piecewise definitions.

• Example: For $f(x) = \begin{cases} 1 & x < 0 \\ 2 & x \geq 0 \end{cases}$, $\lim_{x \to 0^-} f(x) = 1$, $\lim_{x \to 0^+} f(x) = 2$

Limits Involving Infinity

Limits can describe the behavior of functions as x approaches infinity or negative infinity, or as the function itself grows without bound.

• Finite Limits as x Approaches Infinity: $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$

• Limits at Infinity: Used to analyze end behavior of functions.

• Infinite Limits: Occur when the function increases or decreases without bound as x approaches a finite value.

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

Asymptotes of Graphs of Rational Functions

Asymptotes are lines that a graph approaches but never touches. They help describe the behavior of rational functions.

• Horizontal Asymptote: A horizontal line $y = L$ that the graph approaches as $x \to \pm\infty$.

• Vertical Asymptote: A vertical line $x = a$ where the function grows without bound as $x$ approaches $a$.

• Example: For $f(x) = \frac{1}{x}$, $x = 0$ is a vertical asymptote, $y = 0$ is a horizontal asymptote.

Continuity

Continuity describes a function that does not have any abrupt jumps, breaks, or holes at a point or over an interval.

• Definition (Continuity at a Point): A function f(x) is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.

• Continuity Test: Check if the function is defined at the point, the limit exists, and the limit equals the function value.

• Continuous Function: A function is continuous on an interval if it is continuous at every point in the interval.

• Example: $f(x) = x^2$ is continuous everywhere.

Theorems Related to Limits and Continuity

Several important theorems help in evaluating limits and understanding continuity.

• Theorem 1-4: Basic limit laws and properties.

• Theorem 5-6: One-sided limits and endpoint behavior.

• Theorem 7: Infinite limits and vertical asymptotes.

• Theorem 8-9: Criteria for continuity at a point and on intervals.

Summary Table: Types of Limits and Asymptotes

Type	Definition	Example
Finite Limit	Approaches a specific value as x approaches a	$\lim_{x \to 2} (x+1) = 3$
Infinite Limit	Function increases/decreases without bound as x approaches a	$\lim_{x \to 0^+} \frac{1}{x} = \infty$
Limit at Infinity	Function approaches a value as x approaches infinity	$\lim_{x \to \infty} \frac{1}{x} = 0$
Horizontal Asymptote	y-value approached as x goes to infinity	$y = 0$ for $f(x) = \frac{1}{x}$
Vertical Asymptote	x-value where function diverges	$x = 0$ for $f(x) = \frac{1}{x}$

Additional info: The syllabus references specific theorems and textbook pages, indicating a structured approach to teaching limits and continuity, including definitions, properties, and graphical analysis. The inclusion of rational functions and asymptotes suggests coverage of both algebraic and graphical perspectives.