Pre-Calculus Foundations for Calculus: Functions, Limits, and Continuity

Pre-Calculus Review for Calculus

Introduction

Reviewing pre-calculus concepts is essential for a smooth transition into calculus. This section covers foundational topics such as functions, limits, and continuity, which are critical for understanding calculus concepts.

Functions and Their Properties

Domain and Range

• Domain: The set of all possible input values (usually denoted as x) for which a function is defined.

• Range: The set of all possible output values (usually denoted as y) that a function can produce.

• Be able to identify the domain and range for different types of functions.

Function Notation

• Notation: $f(x)$ represents the output of the function f for a given input x.

• Example: If $f(x) = x^2$, then $f(3) = 9$.

Graphs of Elementary Functions

• Know the characteristic shapes of common functions, including:

  • Polynomial functions

  • Rational functions

  • Trigonometric functions

  • Exponential functions

Inverse Functions

• The graph of an inverse function is a reflection of the original function across the line $y = x$.

• Notation: $f^{-1}(x)$ denotes the inverse of $f(x)$.

Trigonometry Review

• Review the unit circle, trigonometric identities, and graphs of trigonometric functions such as:

  • $\sin(x)$ (sine)

  • $\cos(x)$ (cosine)

Limits

Introduction to Limits

The concept of a limit is fundamental in calculus, describing the behavior of a function as its input approaches a certain value.

Limit Notation

• The notation $\lim_{x \to c} f(x) = L$ means that as $x$ gets closer and closer to $c$, $f(x)$ gets closer and closer to $L$.

One-Sided Limits

• Left-hand limit: $\lim_{x \to c^-} f(x)$ describes the behavior of $f(x)$ as $x$ approaches $c$ from the left.

• Right-hand limit: $\lim_{x \to c^+} f(x)$ describes the behavior of $f(x)$ as $x$ approaches $c$ from the right.

• For the general limit $\lim_{x \to c} f(x)$ to exist, both the left-hand and right-hand limits must exist and be equal.

Evaluating Limits

• Direct Substitution: For continuous functions, plug the value into the function.

• Indeterminate Forms: If direct substitution gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, further algebraic manipulation is needed.

• Factoring and Canceling: For rational functions, factor and cancel common terms.

• Conjugates: For expressions with square roots, multiply numerator and denominator by the conjugate.

Limits Involving Infinity

• Infinite Limits: If $f(x)$ grows without bound as $x$ approaches a finite number, there is a vertical asymptote.

• Limits at Infinity: If $f(x)$ approaches a finite number as $x$ becomes very large (positive or negative), there is a horizontal asymptote.

Squeeze Theorem

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

• This theorem is useful for evaluating limits of functions that are difficult to analyze directly.

Continuity

Definition of Continuity at a Point

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

  1. $f(c)$ is defined.

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

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

Types of Discontinuities

• Removable Discontinuity: The limit exists, but the function value does not exist or is different. Graphically, this is a "hole."

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

• Infinite Discontinuity: The limit goes to positive or negative infinity (vertical asymptote).

Continuity on an Interval

• A function $f$ is continuous on a closed interval $[a, b]$ if it is continuous at every point in that interval.

Intermediate Value Theorem

• If $f$ is continuous on $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one $c$ in $[a, b]$ such that $f(c) = N$.

• This theorem is useful for proving the existence of roots or solutions.