Business Calculus: Limits, Continuity, and Differentiation Study Notes

1.1 Limits: A Numerical and Graphical Approach

Limits on a Number Line

Limits describe the value that a function approaches as the input approaches a certain value. Understanding limits is foundational for calculus, as it leads to the concepts of continuity and derivatives.

• Limit from the Left: Approaching a number from values less than the target.

• Limit from the Right: Approaching a number from values greater than the target.

• Two-sided Limit: If both one-sided limits are equal, the two-sided limit exists.

Example: The sequence 0.9, 0.99, 0.999, ... approaches 1 as more 9's are added. We write $\lim_{x \to 1} x = 1$.

Numerical Limits of Functions

Numerical tables can be used to estimate limits by evaluating the function at values close to the target point from both sides.

x	f(x)
0.99	2.98
1.01	3.02

As x approaches 1, f(x) approaches 3, so $\lim_{x \to 1} f(x) = 3$.

Graphical Limits

Limits can also be estimated by examining the graph of a function. If the function approaches the same value from both sides as x approaches a point, the limit exists at that point.

• Draw a vertical line at the target x-value and observe the y-values approached from both sides.

Example: For $f(x) = 2x + 1$, as x approaches 4, $f(x)$ approaches 9.

Theorem: Existence of Limits

• If the left-hand and right-hand limits are equal, the limit exists and equals that value.

• If they are not equal, the limit does not exist.

1.2 Algebraic Limits and Continuity

Algebraic Limits

Limits can often be evaluated using algebraic techniques, especially for polynomials and rational functions.

• Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

• Product Rule: $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$

Example: $\lim_{x \to 7} (-2x^2 + 8x + 1) = -2(7^2) + 8(7) + 1 = -98 + 56 + 1 = -41$

Continuity

A function is continuous at a point if the following three conditions are met:

1. f(a) is defined

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

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

Discontinuities can be classified as jump, removable, or infinite discontinuities.

1.3 Average Rate of Change

Definition

The average rate of change of a function f(x) from $x_1$ to $x_2$ is:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

This measures how much the function's output changes per unit change in input.

Difference Quotient

The difference quotient is a specific form of average rate of change, used to define the derivative:

$\frac{f(x+h) - f(x)}{h}$

As h approaches 0, this expression approaches the derivative of f at x.

1.4 Differentiation Using Limits of Difference Quotients

Tangent Lines

A tangent line touches a curve at exactly one point and has the same slope as the curve at that point.

Definition of the Derivative

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 represents the instantaneous rate of change of the function at $x = a$.

1.5 Differentiation Techniques: The Power and Sum-Difference Rules

Power Rule

For any real number n:

$\frac{d}{dx} x^n = n x^{n-1}$

Sum and Difference Rule

The derivative of a sum (or difference) is the sum (or difference) of the derivatives:

$\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)$

1.6 Differentiation Techniques: The Product and Quotient Rules

Product Rule

If $\sigma(x) = M(x)N(x)$, then:

$\sigma'(x) = M'(x)N(x) + M(x)N'(x)$

Quotient Rule

If $\sigma(x) = \frac{M(x)}{N(x)}$, then:

$\sigma'(x) = \frac{M'(x)N(x) - M(x)N'(x)}{[N(x)]^2}$

Applications

• Used to differentiate average cost, revenue, and profit functions in business applications.

1.7 The Chain Rule

Composition of Functions

The composition $f \circ g$ is defined as $f(g(x))$.

The Chain Rule

If $y = f(g(x))$, then:

$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

1.8 Higher Order Derivatives

Definition

Higher order derivatives are derivatives of derivatives. The second derivative is $f''(x)$, the third is $f'''(x)$, and so on.

Velocity and Acceleration

• Velocity: $v(t) = s'(t)$, the derivative of position with respect to time.

• Acceleration: $a(t) = v'(t) = s''(t)$, the derivative of velocity with respect to time.

Summary Table: Limit Properties

Property	Formula
Sum	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
Product	$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
Quotient	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$
Power	$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$

Additional info: These notes cover foundational calculus concepts essential for business applications, including limits, continuity, average rate of change, and differentiation techniques. Examples and graphical illustrations are provided throughout to reinforce understanding.