Limits and Their Properties in Business Calculus

Limits and Their Properties

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for analyzing continuity, rates of change, and the behavior of functions at boundaries.

• Limit of a function as x approaches a value describes the value that the function approaches.

• Limits can be evaluated as x approaches a finite number or infinity.

Limits at Infinity

When analyzing graphs, it is important to consider what happens to the y-values as x becomes very large (approaches infinity).

• Example A: For the graph of $y = x^2$, as $x \to \infty$, $y \to \infty$.

• Example B: For the graph of $y = \frac{2}{x}$, as $x \to \infty$, $y \to 0$ (horizontal asymptote at $y=0$).

Notation:

• $\lim_{x \to \infty} f(x) = L$

• For Example A: $\lim_{x \to \infty} x^2 = \infty$

• For Example B: $\lim_{x \to \infty} \frac{2}{x} = 0$

One-Sided Limits and Limit Existence

For a limit to exist at $x=a$, both the left-hand and right-hand limits must exist and be equal.

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

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

• If these are not equal, the limit at $x=a$ does not exist.

Example: For a function $f(x)$, if $\lim_{x \to 2^-} f(x) = 0$ and $\lim_{x \to 2^+} f(x) = -1$, then $\lim_{x \to 2} f(x)$ does not exist.

Evaluating Limits from Graphs

Limits can be estimated using the graph of a function by observing the behavior as $x$ approaches a specific value.

• Example: Given a graph, find $\lim_{x \to -2} f(x)$, $\lim_{x \to 0} f(x)$, $f(-1)$, $\lim_{x \to 1} f(x)$, $f(0)$, $\lim_{x \to 2} f(x)$.

• Use the graph to read the y-value as x approaches the specified value from both sides.

• If the left and right limits are not equal, the limit does not exist (DNE).

Properties of Limits

Limits follow several algebraic properties that allow for easier computation.

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

• Constant Rule: $\lim_{x \to a} c = c$

• 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 $\lim_{x \to a} g(x) \neq 0$

Examples of Limit Calculations

• $\lim_{x \to 0} x = 0$

• $\lim_{x \to 0} x^3 = 0$

• $\lim_{x \to \infty} x = \infty$

• $\lim_{x \to \infty} 10 = 10$

• $\lim_{x \to a} (4x + 1) = \lim_{x \to a} 4x + \lim_{x \to a} 1 = 4 \cdot \lim_{x \to a} x + 1$

Table: Limit Existence Criteria

The following table summarizes the conditions for the existence of a limit at a point:

Condition	Description
Left-hand limit exists	$\lim_{x \to a^-} f(x)$ exists
Right-hand limit exists	$\lim_{x \to a^+} f(x)$ exists
Limits are equal	$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$
Limit exists at $x=a$	All above conditions are satisfied

Summary

• Limits describe the behavior of functions as inputs approach specific values.

• For a limit to exist at a point, both one-sided limits must exist and be equal.

• Limits can be evaluated using graphs, algebraic properties, and direct substitution.

• Understanding limits is crucial for further topics in calculus, such as continuity and derivatives.