Limits and Their Properties in Business Calculus

Limits in Calculus

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 other key ideas in business calculus.

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

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

• Example: For $f(x) = x^2$, as $x$ approaches 2, $f(x)$ approaches 4.

Limit Notation and Interpretation

Limit notation can include arrows and signs to indicate direction:

• Left-hand limit: $\lim_{x \to a^-} f(x)$ approaches $a$ from values less than $a$.

• Right-hand limit: $\lim_{x \to a^+} f(x)$ approaches $a$ from values greater than $a$.

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

Estimating Limits Using Tables and Graphs

Limits can be estimated by evaluating function values near the point of interest or by examining the graph.

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	?	?	?	?	?	?

As $x$ approaches 2, $f(x) = x^2$ approaches 4.

Limit of a Rational Function

Rational functions may be undefined at certain points, but limits can still exist.

x	1.9	1.99	1.999	2.001	2.01	2.1
g(x)	?	?	Undefined	?	?	?

For $g(x) = \frac{x^2 - 2x}{x - 2}$, the function is undefined at $x = 2$, but the limit as $x \to 2$ can be found by simplifying or using a table.

Limits of Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Limits at points where the definition changes require checking both sides.

x	1.9	1.99	1.999	2.001	2.01	2.1
h(x)	2	2	2	1	1	1

For $h(x) = \begin{cases} x^2, & x \neq 2 \\ 1, & x = 2 \end{cases}$, the limit as $x \to 2$ is 4, not 1.

Existence of Limits

Conditions for Limit Existence

A limit exists at $x = a$ if the left-hand and right-hand limits are equal and finite. If $f(x)$ becomes infinitely large or the one-sided limits differ, the limit does not exist.

• If $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$, then $\lim_{x \to a} f(x)$ does not exist.

• If $f(x)$ approaches $\infty$ or $-\infty$ as $x$ approaches $a$, the limit does not exist.

Rules for Limits

Constant Rule

If $k$ is a constant, then:

• $\lim_{x \to a} k = k$

Sum or Difference Rule

The limit of a sum or difference is the sum or difference of the limits:

• $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$

Product Rule

The limit of a product is the product of the limits:

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

Quotient Rule

The limit of a quotient is the quotient of the limits, provided the denominator is not zero:

• $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$

Power Rule

For any real number $k$:

• $\lim_{x \to a} [f(x)]^k = [\lim_{x \to a} f(x)]^k$

Equality Rule

If $f(x) = g(x)$ for all $x$ near $a$, then:

• $\lim_{x \to a} f(x) = \lim_{x \to a} g(x)$

Exponent and Logarithm Rules

• Exponent Rule: For $b > 0$, $\lim_{x \to a} b^{f(x)} = b^{\lim_{x \to a} f(x)}$

• Logarithm Rule: For $A > 0$, $\lim_{x \to a} \log_b f(x) = \log_b (\lim_{x \to a} f(x))$

Limits at Infinity

Limits as $x$ Approaches Infinity

Limits at infinity describe the behavior of functions as $x$ becomes very large or very small.

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

• Example: $\lim_{x \to -\infty} \frac{1}{x} = 0$

x	1000	100	10	1	-1	-10	-100	-1000
$f(x) = \frac{1}{x}$	0.001	0.01	0.1	1	-1	-0.1	-0.01	-0.001

Limits at Infinity for Rational Functions

For rational functions, the degree of the numerator and denominator determines the limit at infinity.

• If degree of denominator > degree of numerator: Limit is 0.

• If degrees are equal: Limit is the ratio of leading coefficients.

• If degree of numerator > degree of denominator: Limit is $\infty$ or $-\infty$.

x	1000	100	10	1	-1	-10	-100	-1000
$f(x) = \frac{1}{x^2}$	0.000001	0.0001	0.01	1	1	0.01	0.0001	0.000001

Applications: Oxygen Concentration Example

Modeling with Limits

Limits are used in business and science to model long-term behavior. For example, the concentration of oxygen in a pond after pollution can be modeled by:

• $f(t) = \frac{12t^2 + 15t + 12}{t^2 + t + 1}$

• As $t \to \infty$, $f(t)$ approaches 12.

t	2	4	10	100	1,000	10,000	100,000
$f(t)$	?	?	?	?	?	?	?

Summary Table: Limit Rules

Rule	Formula	Example
Constant Rule	$\lim_{x \to a} k = k$	$\lim_{x \to 3} 5 = 5$
Sum/Difference Rule	$\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$	$\lim_{x \to 2} (x^2 + 3x) = 4 + 6 = 10$
Product Rule	$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$	$\lim_{x \to 2} (x \cdot x^2) = 2 \cdot 4 = 8$
Quotient Rule	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$	$\lim_{x \to 2} \frac{x^2}{x} = \frac{4}{2} = 2$
Power Rule	$\lim_{x \to a} [f(x)]^k = [\lim_{x \to a} f(x)]^k$	$\lim_{x \to 2} (x^2)^3 = 4^3 = 64$
Exponent Rule	$\lim_{x \to a} b^{f(x)} = b^{\lim_{x \to a} f(x)}$	$\lim_{x \to 2} 2^{x} = 2^2 = 4$
Logarithm Rule	$\lim_{x \to a} \log_b f(x) = \log_b (\lim_{x \to a} f(x))$	$\lim_{x \to 2} \log_2 x = \log_2 2 = 1$

Additional info:

• Limits are used in business calculus to analyze marginal cost, marginal revenue, and other rates of change.

• Piecewise functions often model real-world scenarios with abrupt changes, such as tax brackets or shipping rates.

• Limits at infinity help determine long-term trends and equilibrium values in business models.