Rates of Change and Limits Study Notes

Section 2.1 Rates of Change and Limits

Theorem 1: Properties of Limits

• If $L$, $M$, $c$, and $k$ are real numbers and propertie limx→c f(x) = L and limx→c g(x) = M, then:

  1. Sum Rule:
     $\lim_{x \to c} (f(x) + g(x)) = L + M$

  • The limit of the sum of two functions is the sum of their limits.

  1. Difference Rule:
     $\lim_{x \to c} (f(x) - g(x)) = L - M$

  • The limit of the difference of two functions is the difference of their limits.

  1. Product Rule:
     $\lim_{x \to c} (f(x) \cdot g(x)) = L \cdot M$

  • The limit of a product of two functions is the product of their limits.

  1. Constant Multiple Rule:
     $\lim_{x \to c} (k \cdot f(x)) = k \cdot L$

  • The limit of a constant times a function is the constant times the limit of the function.

  1. Quotient Rule:
     $\lim_{x \to c} \left(\frac{f(x)}{g(x)}\right) = \frac{L}{M}, \text{ provided } M \neq 0$

  • The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.

Properties of Limits

• By applying six basic facts about limits, we can calculate many unknown limits from limits we already know:

  • For instance, knowing that $\lim{x \to c} k = k$ (the limit of the constant function $k$) and $\lim{x \to c} x = c$ (the limit of the identity function at $x = c$), we can calculate the limits of all polynomial and rational functions.

Example 3: Using Properties of Limits

• Use the observations of limits to find the following:

  • (a) $\lim_{x \to c} (x^3 - 4x^2 + 3)$:

  • Applying the rules:

  • $\lim<em>{x \to c} (x^3) = c^3,$ $\lim</em>{x \to c} (4x^2) = 4c^2,$ $\lim_{x \to c} (3) = 3$

  • Thus, $\lim_{x \to c} (x^3 - 4x^2 + 3) = c^3 - 4c^2 + 3$

  • (b) $\lim_{x \to c} \left(\frac{x^4 - x^2}{x^2 - 5}\right)$:

  • $\lim<em>{x \to c}(x^4) = c^4,$ $\lim</em>{x \to c}(x^2) = c^2,$ $\lim_{x \to c}(5) = 5$

  • Hence, $\lim_{x \to c} \left(\frac{x^4 - x^2}{x^2 - 5}\right) = \frac{c^{4} - c^{2}}{c^{2} - 5}$

Theorem 2: Polynomial and Rational Functions

1. If $f(x) = an x^n + a{n-1} x^{n-1} + … + a0$ is any polynomial function and $c$ is any real number, then: $\lim</em>{x \to c} f(x) = f(c) = a<em>n c^n + a</em>{n-1} c^{n-1} + … + a_0$

2. If $f(x)$ and $g(x)$ are polynomials and $c$ is any real number, then:
   $\lim_{x \to c} \left(\frac{f(x)}{g(x)}\right) = \frac{f(c)}{g(c)}$, provided $g(c) \neq 0$.

Example 4: Using Theorem 2

(a) $\lim{x \to 3} (x^2 - x)$: $= 3^2 - 3 = 6$ (b) $\lim{x \to 2} (2x^2 + 5)$:
$= 2(2^2) + 5 = 13$

Example 5: Using the Product Rule

• Determine $\lim_{x \to 0} \left(\frac{\tan x}{x}\right)$:

  • Graphical approach suggests the limit approaches 1. Analytical approach using:
    $\lim<em>{x \to 0} \left(\frac{\tan x}{x}\right) = \lim</em>{x \to 0} \left(\sin x / x \cdot \cos x \right) = 1 \cdot 1 = 1$

Example 6: Exploring a Nonexistent Limit

• Graphically show that

  • $\lim_{x \to 2} (\frac{x^3 - 1}{x-2})$ does not exist.
    This expresses that the function values approach infinity, suggesting the limit does not exist.

The One-sided and Two-sided Limits

• The limit of a function $f$ as $x$ approaches a number $c$ may vary depending on whether $x$ approaches $c$ from the left or the right.

• Right-hand limit:
  $\lim_{x \to c^+} f(x)$

• Left-hand limit:
  $\lim_{x \to c^-} f(x)$

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

Example 7: Values Approaching Two Numbers

• The greatest integer function $f(x) = int(x)$ shows different limits at each integer.

  • E.g., $\lim{x \to 3} int(x) = 3$ from the right, and $\lim{x \to 3} int(x) = 2$ from the left.

Theorem 3: One-sided and Two-sided Limits

• A function $f(x)$ has a limit as $x$ approaches $c$ iff both right-hand and left-hand limits exist and are equal.

Infinite Limits as $x \to a$

• If the values of a function $f(x)$ exceed all positive (or negative) bounds as $x$ approaches a finite number $a$, then:

  • $\lim<em>{x \to a} f(x) = \infty$ or $\lim</em>{x \to a} f(x) = -\infty$

• Example: $f(x) = \frac{1}{x}$ has vertical asymptotes at 0.

Sandwich Theorem

• If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $c$, and:

  • $\lim<em>{x \to c} g(x) = L ext{ and } \lim</em>{x \to c} h(x) = L$
    then $\lim_{x \to c} f(x) = L$.

Example 9: Using the Sandwich Theorem

• $\lim_{x \to 0} (x^2 \sin(1/x)) = 0.$

• The values of $ ext{sin}(1/x)$ are bounded between -1 and 1, leading us to express:

  • $x^2(-1) \leq x^2 \sin(1/x) \leq x^2(1)$

  • Limits confirm this.

Quick Review 2.1

Exercises

• Exercises exploring limits and functions' expressions suggested for practice and solidification of concepts discussed.

Exercise Topics

• 1-4: Finding limits, simple polynomials.

• 5-8: Working with inequalities in limit processes.

• 9-12: Reducing fractions to find limits.

• 13-18: Verifying limits through substitution and graphical validation.

• 19-26: Applying limit definitions to complex functions.

• 27-30: Comprehending graphs for limit conditions.

Concluding Concepts

• The understanding of limits underpins calculus, providing foundational tools to interpret continuous change, often represented in real-world applications.

• Maintain knowledge of diverse rules and theorems to effectively navigate complex limit scenarios.