Calculus Study Notes - Limits and Functions chapter 10

Introduction

• The discussion focuses on the application of limits in business and economics courses rather than abstract mathematical details.

Understanding Limits

Basic Concepts

• Definition of limit:

  • The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$.

  • Example: If $f(x) = x$, then $\lim_{{x \to a}} f(x) = a$ since as $x$ approaches $a$, $f(x)$ gets closer to $a$.

• Limits of constant functions:

  • If $f(x) = c$ (constant function), then $\lim_{{x \to a}} f(x) = c$ since $c$ does not change regardless of $x$.

Limit Examples

• Examples of limits:

  • $\lim_{{x \to 3}} f(x) = 3$

  • $\lim_{{x \to 2}} f(x) = 2$

  • $\lim_{{x \to 3}} 15 = 15$

Properties of Limits

Properties of Addition and Subtraction

1. Limit of a Sum:

   • If $f(x)$ and $g(x)$ are functions, then:
     $\lim<em>{{x \to a}} [f(x) + g(x)] = \lim</em>{{x \to a}} f(x) + \lim_{{x \to a}} g(x)$

2. Limit of a Difference:

   • If $f(x)$ and $g(x)$ are functions, then:
     $\lim<em>{{x \to a}} [f(x) - g(x)] = \lim</em>{{x \to a}} f(x) - \lim_{{x \to a}} g(x)$

Properties of Multiplication

3. Limit of a Product:

   • If $f(x)$ and $g(x)$ are functions, then:
     $\lim<em>{{x \to a}} [f(x) \cdot g(x)] = \lim</em>{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)$

Properties of Division

4. Limit of a Quotient:

   • If $f(x)$ and $g(x)$ are functions, then:
     $\lim<em>{{x \to a}} \left[ \frac{f(x)}{g(x)} \right] = \frac{\lim</em>{{x \to a}} f(x)}{\lim<em>{{x \to a}} g(x)}$ provided that $\lim</em>{{x \to a}} g(x) \neq 0$.

• Remarks: Violating the condition of non-zero denominators leads to undefined limits.

Direct Substitution Property

• A function $f(x)$ has the direct substitution property if:

  • For rational functions, direct substitution can be used to evaluate limits at points in the domain (where the denominator is not zero).

  • Polynomial functions: Use direct substitution to find limits.

  • Example: For $f(x) = 5x^2 + 2x + 3$, to find $\lim_{{x \to 2}} f(x)$, substitute $x=2$:

  • $5(2^2) + 2(2) + 3 = 20 + 4 + 3 = 27$

Rational Functions

• Definition: A rational function is expressed as $\frac{p(x)}{q(x)}$ where $p$ and $q$ are polynomials.

• Direct substitution property applies if $q(a) \neq 0$.

• Example: To compute $\lim_{{x \to 3}} \frac{x^3 - x^2 + 1}{3x^2 - 3}$, substitute $x=3$:

  • $\frac{27 - 9 + 1}{27 - 3} = \frac{19}{24}$

Functions with Direct Substitution Property

Types of Functions with Direct Substitution

• Functions that typically allow for direct substitution include:

  • Polynomial functions

  • Rational functions

  • Radical functions (all roots)

  • Exponential functions (e.g., $2^x$, $e^x$)

  • Logarithmic functions

  • Trigonometric functions

  • Inverse trigonometric functions

Importance of Domain

• Ensure the value approached does not cause a division by zero; if it does, alternative methods of evaluation are needed (factoring, conjugates, etc.)

Evaluating Limits - Practices

Example Evaluations

• Example 1: $\lim_{{x \to 1}} \frac{x + 1}{x + 2}$

  • Direct substitution gives:

  • $\frac{2}{3}$ (works)

• Example 2: $\lim_{{x \to 2}} \frac{x + 4}{x - 2}$

  • Direct substitution gives division by zero (6/0), fails.

• Recognize form: nonzero constant/zero denotes future evaluation needs.

Example with Factoring

• Evaluate $\lim_{{x \to -2}} \frac{x^{2} - 1}{x + 2}$.

• Direct substitution gives 0/0 (fail).

• Factor and cancel terms before re-evaluating.

Techniques for Zero Over Zero Forms

1. Factoring

2. Multiplication by Conjugate

3. L'Hôpital's Rule (for advanced courses, not covered in this context).

Final Remarks on Evaluating Limits

• Direct substitution should be attempted initially; if it fails, apply algebraic methods to manipulate the expression, then re-evaluate using substitution.

• Keep notation clear and maintain aware of limits until evaluation is complete.

• Understand distinctions between definitions and practical computation in calculus to avoid errors.