Algebraic Limits, Average Rates of Change, and Differentiation in Business Calculus

1.2 Algebraic Limits

Introduction to Limits

Limits are foundational in calculus, describing the behavior of functions as inputs approach a particular value. Understanding limits is essential for analyzing continuity, rates of change, and derivatives.

• Limit Notation: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

• Limits can be evaluated from the left or right, and the value must be the same from both sides for the limit to exist.

Techniques for Finding Limits

There are three main algebraic techniques for evaluating limits: direct substitution, cancellation, and rationalization.

Direct Substitution

• If $f(x)$ is a polynomial or rational function and $a$ is in the domain, then $\lim_{x \to a} f(x) = f(a)$.

• Example: $\lim_{x \to 2} 3x^2 + 4x - 5 = 3(2)^2 + 4(2) - 5 = 12 + 8 - 5 = 15$

• If direct substitution gives an indeterminate form (like $0/0$), use another technique.

Cancellation Technique

• Used when direct substitution gives $0/0$ due to a common factor in numerator and denominator.

• Factor and cancel, then substitute again.

• Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ Factor numerator: $\frac{(x-2)(x+2)}{x-2}$ Cancel $(x-2)$: $\lim_{x \to 2} x+2 = 4$

Rationalization Technique

• Used when limits involve radicals and direct substitution gives $0/0$.

• Multiply numerator and denominator by the conjugate to simplify.

• Example: $\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x}$ Multiply by $\frac{\sqrt{x+1} + 1}{\sqrt{x+1} + 1}$: $\frac{(\sqrt{x+1} - 1)(\sqrt{x+1} + 1)}{x(\sqrt{x+1} + 1)} = \frac{x}{x(\sqrt{x+1} + 1)} = \frac{1}{\sqrt{x+1} + 1}$ Substitute $x = 0$: $\frac{1}{2}$

1.3 Average Rates of Change (AROC)

Definition and Interpretation

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. In business, this can represent average velocity, revenue change, or other rates.

• Formula: $\text{Slope} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• This is the slope of the secant line between points $(x_1, f(x_1))$ and $(x_2, f(x_2))$.

Average Velocity

• Measures how far an object travels per unit time over an interval.

• Formula: $\text{Average Velocity} = \frac{\text{Distance Traveled}}{\text{Time Elapsed}}$

• Example: If you travel 50 miles in 1 hour, average velocity is $50$ mph.

Difference Quotient

• The difference quotient measures the average rate of change over an interval $[x, x+h]$:

$\text{Difference Quotient} = \frac{f(x+h) - f(x)}{h}$

• This is the slope of the secant line between $x$ and $x+h$.

• Example: For $f(x) = x^2$, the difference quotient is $\frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$.

Applications: Average Revenue

• In business, the average rate of change can represent average revenue per unit sold over an interval.

• Example: If $R(x) = 0.001x^3 + 150x$, then $\text{Average Revenue} = \frac{R(305) - R(300)}{305 - 300}$

1.4 Differentiation Using Limits of Difference Quotients

Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change, or the slope of the tangent line at that point. It is defined as the limit of the difference quotient as $h \to 0$.

$\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2$, $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = 2x$

Instantaneous Velocity

• The instantaneous velocity at time $t$ is the derivative of the position function $s(t)$:

$\text{Instantaneous Velocity} = s'(t) = \lim_{h \to 0} \frac{s(t+h) - s(t)}{h}$

• Example: If $s(t) = 5t^2 - 0.8t$, then $s'(t) = 10t - 0.8$

Finding the Equation of the Tangent Line

• The tangent line at $x=a$ has slope $f'(a)$ and passes through $(a, f(a))$.

• Point-slope form: $y - f(a) = f'(a)(x - a)$

When is a Function Not Differentiable?

• If $f(x)$ is not defined at $x=a$

• If $f(x)$ has a discontinuity at $x=a$

• If $f(x)$ has a corner, cusp, or vertical tangent at $x=a$

Example: $f(x) = |x|$ is not differentiable at $x=0$ due to a corner.

Summary Table: Techniques for Finding Limits

Technique	When to Use	Example
Direct Substitution	Function is continuous at the point	$\lim_{x \to 2} x^2 + 3 = 7$
Cancellation	Indeterminate form $0/0$ due to common factor	$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$
Rationalization	Indeterminate form $0/0$ involving radicals	$\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x} = \frac{1}{2}$

Additional info: These notes provide foundational techniques and examples for limits, average rates of change, and differentiation, which are essential for further study in Business Calculus, including applications to business contexts such as revenue and velocity.