Limits and Limit Rules in Business Calculus

Limits and One-Sided Limits

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for analyzing continuity, derivatives, and the behavior of functions near points of interest.

• Limit (lim): The value that a function approaches as the input approaches a specific point.

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

• Does Not Exist (DNE): If the function does not approach a single value, the limit does not exist.

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a value from only one side (left or right).

• Right-hand limit: $\lim_{x \to a^+} f(x)$ (as $x$ approaches $a$ from the right)

• Left-hand limit: $\lim_{x \to a^-} f(x)$ (as $x$ approaches $a$ from the left)

• If the left and right limits are not equal, the two-sided limit does not exist.

Example: If $f(x)$ approaches different values from the left and right at $x=0$, then $\lim_{x \to 0} f(x)$ DNE.

Graphical Interpretation

• Limits can be visualized on a graph by observing the $y$-value that $f(x)$ approaches as $x$ nears a specific point.

• Discontinuities, jumps, or holes in the graph can indicate where limits may not exist.

Limit Rules

Basic Limit Properties

Several algebraic rules allow the computation of limits for combinations of functions, provided the individual limits exist.

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

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

• Constant Multiple Rule: $\lim_{x \to a} [k \cdot f(x)] = k \cdot \lim_{x \to a} f(x)$, where $k$ is a constant

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

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

Example: $\lim_{x \to 2} (3x^2 + 2x - 5) = 3(2^2) + 2(2) - 5 = 12 + 4 - 5 = 11$

Limits of Polynomials and Rational Functions

• If $P(x)$ and $Q(x)$ are polynomials, then $\lim_{x \to a} P(x) = P(a)$ and $\lim_{x \to a} \frac{P(x)}{Q(x)} = \frac{P(a)}{Q(a)}$ (provided $Q(a) \neq 0$).

• If substitution leads to $\frac{0}{0}$ (an indeterminate form), algebraic manipulation is required to evaluate the limit.

Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ gives $\frac{0}{0}$, so factor numerator: $\frac{(x-2)(x+2)}{x-2} = x+2$, so limit is $4$.

Special Cases and Theorems

• If $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, the limit $\lim_{x \to a} \frac{f(x)}{g(x)}$ may be indeterminate ($\frac{0}{0}$ form).

• Algebraic simplification (factoring, canceling) is often necessary to resolve indeterminate forms.

Example: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x-3)(x+3)}{x-3} = 6$

Absolute Value and Limits

• Limits involving absolute value may require considering one-sided limits.

• If the left and right limits are not equal, the two-sided limit does not exist.

Example: $\lim_{x \to 0} \frac{|x|}{x}$ does not exist because the left and right limits are $-1$ and $1$, respectively.

Summary Table: Limit Rules

Rule	Formula	Conditions
Sum	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$	Both limits exist
Difference	$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$	Both limits exist
Constant Multiple	$\lim_{x \to a} [k \cdot f(x)] = k \cdot \lim_{x \to a} f(x)$	$k$ is a constant
Product	$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$	Both limits exist
Quotient	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$	$\lim_{x \to a} g(x) \neq 0$

Key Takeaways

• Limits describe the behavior of functions near specific points and are essential for calculus.

• One-sided limits help analyze discontinuities and piecewise functions.

• Algebraic rules simplify the computation of limits for sums, differences, products, and quotients.

• Special attention is needed for indeterminate forms and absolute value functions.