Limits and One-Sided Limits: Foundations for Business Calculus

Limits and One-Sided Limits

Introduction to Limits

The concept of a limit is fundamental in calculus and is used to describe the behavior of a function as its input approaches a particular value. Understanding limits is essential for analyzing functions, especially when direct evaluation is not possible due to indeterminate forms.

• Definition: The limit of a function f(x) as x approaches a is denoted by $\lim_{x \to a} f(x) = L$, where L is a unique real number that f(x) approaches as x gets close to a (but not equal to a).

• Indeterminate Forms: Sometimes, direct substitution leads to expressions like $\frac{0}{0}$, which require further analysis.

• Numerical Approximation: Limits can be approximated by evaluating the function at values increasingly close to the target point.

Example: Numerical Approach to Limits

Consider $f(x) = \frac{x^2 - 4}{x - 2}$. Direct substitution at $x = 2$ gives $\frac{0}{0}$, an indeterminate form. By evaluating $f(x)$ for values near 2, we observe:

x	f(x)
1.9	3.9
1.99	3.99
1.999	3.999
1.9999	3.9999
2	?
2.0001	4.0001
2.001	4.001
2.01	4.01
2.1	4.1

As x approaches 2, f(x) approaches 4. Thus, $\lim_{x \to 2} f(x) = 4$.

One-Sided Limits

Limits can be considered from either side of the target value:

• Left-Hand Limit: $\lim_{x \to a^-} f(x)$, as x approaches a from values less than a.

• Right-Hand Limit: $\lim_{x \to a^+} f(x)$, as x approaches a from values greater than a.

• Existence of Limit: The limit $\lim_{x \to a} f(x)$ exists only if both one-sided limits are equal.

Piecewise Functions and Limits

Piecewise functions may have different expressions on either side of a point, affecting the limit at that point.

• Example: $g(x) = \begin{cases} 1 + 2x & \text{if } x \leq 1 \\ 2 - x & \text{if } x > 1 \end{cases}$

• To find $\lim_{x \to 1^-} g(x)$, use the first expression; for $\lim_{x \to 1^+} g(x)$, use the second.

• If the two one-sided limits differ, $\lim_{x \to 1} g(x)$ does not exist.

Limits from Graphs

Limits can be estimated visually by observing the behavior of a function's graph near the point of interest.

• Key Steps:

  1. Identify the value the function approaches from the left and right.

  2. Check for jumps, holes, or asymptotes.

  3. Compare one-sided limits to determine if the overall limit exists.

• Example: If the graph approaches 2 from both sides as x approaches 0, then $\lim_{x \to 0} f(x) = 2$.

Limits Involving Infinity

Some functions do not approach a finite value as x approaches a certain point, but instead increase or decrease without bound.

• Infinite Limits: $\lim_{x \to a} f(x) = \infty$ or $-\infty$ indicates the function grows arbitrarily large or small near a.

• Example: $f(x) = \frac{1}{x^2}$ as $x \to 0$ approaches $\infty$ from both sides.

• Limits at Infinity: $\lim_{x \to \infty} f(x)$ describes the behavior as x becomes very large.

• Example: $\lim_{x \to \infty} \frac{1}{x^2} = 0$.

Numerical and Graphical Practice

Practice problems involve filling tables with values of f(x) near the point of interest, or interpreting graphs to estimate limits.

• Numerical Tables: Evaluate f(x) for values approaching the target from both sides.

• Graphical Estimation: Observe the function's approach to a value or infinity as x nears the point.

Summary Table: Types of Limits

Type	Notation	Description
Two-sided limit	$\lim_{x \to a} f(x)$	Approach from both sides
Left-hand limit	$\lim_{x \to a^-} f(x)$	Approach from left
Right-hand limit	$\lim_{x \to a^+} f(x)$	Approach from right
Infinite limit	$\lim_{x \to a} f(x) = \infty$	Function grows without bound
Limit at infinity	$\lim_{x \to \infty} f(x)$	Behavior as x becomes large

Key Formulas and Notations

• $\lim_{x \to a} f(x) = L$

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

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

• $\lim_{x \to \infty} f(x)$ (limit at infinity)

• $\lim_{x \to a} f(x) = \infty$ (infinite limit)

Applications in Business Calculus

Limits are used to analyze cost, revenue, and profit functions, especially when modeling marginal changes or discontinuities. Understanding limits is foundational for later topics such as differentiation and integration.

• Marginal Analysis: Limits help define marginal cost and marginal revenue as instantaneous rates of change.

• Optimization: Limits are used to find maximum and minimum values in business contexts.

Practice Example

Example: Approximate $\lim_{x \to 3} (2x - 1)$ numerically:

x	f(x)
2.9	4.8
2.99	4.98
2.999	4.998
2.9999	4.9998
3	5
3.0001	5.0002
3.001	5.002
3.01	5.02
3.1	5.2

As x approaches 3, f(x) approaches 5.

Additional info: The notes cover foundational limit concepts, including numerical and graphical approaches, one-sided limits, infinite limits, and their applications. These are essential for Business Calculus and provide the groundwork for differentiation and integration.