Limits: A Numerical and Graphical Approach (Business Calculus Study Notes)

Limits: A Numerical and Graphical Approach

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, derivatives, and applications in business calculus.

• Limit of a Function: The value that a function approaches as the input approaches a certain point.

• Notation: The limit of f(x) as x approaches a is written as $\lim_{x \to a} f(x) = L$.

• Existence: A limit exists if the function approaches the same value from both the left and the right of a.

Numerical and Graphical Methods for Finding Limits

Limits can be estimated using tables of values (numerical approach) or by analyzing the graph of a function (graphical approach).

• Numerical Approach: Construct a table of values for x approaching a from both sides and observe the trend in f(x).

• Graphical Approach: Observe the behavior of the function's graph as x approaches a from the left and right.

Sequences and Limits

Sequences can illustrate how values approach a limit. Consider the following examples:

• Example 1a: Sequence: 2.24, 2.249, 2.2499, 2.24999, ... Limit: Approaches 2.25 from the left ($x \to 2.25^-$).

• Example 1b: Sequence: 5.51, 5.501, 5.5001, 5.50001, ... Limit: Approaches 5.5 from the right ($x \to 5.5^+$).

• Example 1c: Sequence: $\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \frac{31}{32}, \frac{63}{64}, ...$ Limit: Approaches 1 from the left ($x \to 1^-$).

Formal Definition of a Limit

The limit of a function f(x) as x approaches a is L if the values of f(x) get arbitrarily close to L as x gets sufficiently close to a (but not equal to a).

• Mathematical Notation: $\lim_{x \to a} f(x) = L$

Theorem: Existence of a Limit

A limit exists at x = a if and only if the left-hand and right-hand limits both exist and are equal.

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

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

• If both are equal: $\lim_{x \to a} f(x) = L$

Worked Examples

Example 2: Evaluating a Limit with an Indeterminate Form

Given $f(x) = \frac{x^2 - 9}{x - 3}$, find $f(3)$ and $\lim_{x \to 3} f(x)$.

• Direct Substitution: $f(3) = \frac{3^2 - 9}{3 - 3} = \frac{0}{0}$ (indeterminate form; $f(3)$ does not exist).

• Numerical Approach: Use tables to approach $x = 3$ from both sides.

x → 3⁻	2	2.5	2.9	2.99	2.999
f(x)	5	5.5	5.9	5.99	5.999
x → 3⁺	4	3.5	3.1	3.01	3.001
f(x)	7	6.5	6.1	6.01	6.001

• Conclusion: Both sides approach 6, so $\lim_{x \to 3} f(x) = 6$.

Example 3: Piecewise Function and Limits

Given $H(x) = \begin{cases} 2x + 2 & x < 1 \\ 2x - 4 & x \geq 1 \end{cases}$, find $\lim_{x \to 1} H(x)$ and $\lim_{x \to -3} H(x)$.

• At $x = 1$: Left-hand limit: $2(1) + 2 = 4$; Right-hand limit: $2(1) - 4 = -2$

• Conclusion: $\lim_{x \to 1^-} H(x) = 4$, $\lim_{x \to 1^+} H(x) = -2$; limits are not equal, so $\lim_{x \to 1} H(x)$ does not exist.

• At $x = -3$: Both sides use $2x + 2$ (since $x < 1$), so $H(-3) = 2(-3) + 2 = -4$; both left and right limits are $-4$.

• Conclusion: $\lim_{x \to -3} H(x) = -4$.

Left-Hand and Right-Hand Limits

When evaluating limits, it is important to consider the direction from which x approaches a:

• 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)

• If left and right limits are not equal, the limit does not exist at that point.

Graphical Interpretation and the "Wall" Method

The "Wall" Method is a visual technique for determining limits on a graph:

• Draw a vertical line (the "wall") at x = a.

• Trace the curve from the left and right up to the wall.

• If both traces meet at the same height, the limit exists and equals that value.

• If not, the limit does not exist at x = a.

Limits Involving Infinity

Some limits involve x approaching infinity or negative infinity, or the function approaching infinity.

• Notation: $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$

• Interpretation: Describes the end behavior of the function.

Summary Table: Limit Existence Criteria

Situation	Left-Hand Limit	Right-Hand Limit	Limit Exists?
Both limits equal L	L	L	Yes, $\lim_{x \to a} f(x) = L$
Limits not equal	L1	L2	No
One or both limits do not exist	Does not exist	Does not exist	No

Key Takeaways

• Limits describe the behavior of functions near specific points.

• Numerical and graphical methods are practical tools for estimating limits.

• The limit exists at a point if and only if the left-hand and right-hand limits are equal.

• The value of the function at a point does not necessarily equal the limit at that point.

• Tables and graphs are essential for visualizing and understanding limits in business calculus.

Additional info: In business calculus, limits are used to analyze marginal cost, marginal revenue, and instantaneous rates of change, which are foundational for understanding derivatives and optimization problems.