Limits and Continuity: Business Calculus Study Notes

Limits and Continuity

Computing Limits Algebraically

Limits are fundamental to calculus, describing the behavior of functions as inputs approach specific values. Several properties allow us to compute limits algebraically.

• Constant Rule: The limit of a constant is the constant itself. $\lim_{x \to a} c = c$

• Power Rule: The limit of a function raised to a power equals the power of the limit. $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$

• Root Rule: The limit of a root equals the root of the limit. $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$

• Sum/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's limit 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 $\lim_{x \to a} g(x) \neq 0$

• Constant Multiple Rule: The limit of a constant times a function is the constant times the limit. $\lim_{x \to a} c \cdot f(x) = c \cdot \lim_{x \to a} f(x)$

Example:

• Compute $\lim_{x \to 2} (x^2 - x + 3)$: Substitute $x = 2$: $2^2 - 2 + 3 = 4 - 2 + 3 = 5$.

• Compute $\lim_{x \to 1} \frac{x + 3}{x^2 + 1}$: Substitute $x = 1$: $\frac{1 + 3}{1^2 + 1} = \frac{4}{2} = 2$.

• Compute $\lim_{x \to 0} \sqrt{x + 4}$: Substitute $x = 0$: $\sqrt{0 + 4} = 2$.

Limits Involving Indeterminate Forms

Sometimes, direct substitution yields an indeterminate form such as $\frac{0}{0}$. In these cases, algebraic manipulation (like factoring) is necessary.

• Factoring: Factor numerator and denominator to simplify the expression and remove the indeterminate form.

Example:

• Compute $\lim_{x \to 2} \frac{x^2 + 3x - 10}{x^2 - 4}$: Factor numerator: $x^2 + 3x - 10 = (x + 5)(x - 2)$ Factor denominator: $x^2 - 4 = (x + 2)(x - 2)$ Cancel $(x - 2)$: $\frac{x + 5}{x + 2}$ Substitute $x = 2$: $\frac{2 + 5}{2 + 2} = \frac{7}{4}$

Limits at Infinity

Limits as $x$ approaches infinity describe the end behavior of functions. For rational functions, the degree of the numerator and denominator determines the limit.

• Basic Rule: $\lim_{x \to \infty} \frac{1}{x} = 0$ and $\lim_{x \to -\infty} \frac{1}{x} = 0$

• General Rule: If $c$ is a constant and $n$ is a positive integer: $\lim_{x \to \pm\infty} \frac{c}{x^n} = 0$

Example:

• Compute $\lim_{x \to \infty} \frac{x}{x + 1}$: Divide numerator and denominator by $x$: $\frac{1}{1 + \frac{1}{x}}$ As $x \to \infty$, $\frac{1}{x} \to 0$, so limit is $1$.

• Compute $\lim_{x \to \infty} \frac{x + 1}{x^2 + 3x + 1}$: Highest degree in denominator is $x^2$, so limit is $0$.

Definition of Continuity

Continuity describes whether a function has any breaks, jumps, or holes at a point or over an interval.

• Continuous at a Point: A function $f$ is continuous at $a$ if:

  1. $f(a)$ is defined

  2. $\lim_{x \to a} f(x)$ exists

  3. $\lim_{x \to a} f(x) = f(a)$

• Discontinuity: If any of the above conditions fail, $f$ has a discontinuity at $a$.

• Continuous on an Interval: $f$ is continuous on an open interval $I$ if it is continuous at every point in $I$.

• Everywhere Continuous: $f$ is continuous on $(-\infty, \infty)$.

Example:

• Function $f(x) = \frac{x^2 - 4}{x - 2}$ is not defined at $x = 2$ (division by zero), but for $x \neq 2$, $f(x) = x + 2$. The limit as $x \to 2$ is $4$, but$f(2)$is undefined, so$f$is not continuous at$x = 2$.

Types of Discontinuity

Discontinuities can be classified as follows:

Type	Description	Example
Removable	Hole in the graph; limit exists, but function is not defined or not equal to limit at that point.	$f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$
Jump	Function jumps from one value to another; left and right limits exist but are not equal.	Piecewise function with different values at a point
Infinite	Function approaches infinity at a point; limit does not exist.	$f(x) = \frac{1}{x}$ at $x = 0$

Continuity of Piecewise Functions

For piecewise functions, check continuity at the points where the formula changes.

• Evaluate left and right limits at the transition point.

• Check if the function value matches the limit.

Example:

• Let $f(x) = \begin{cases} \frac{1}{3x + 4}, & x \leq 3 \\ 2x - 1, & x > 3 \end{cases}$ Check continuity at $x = 3$: Left limit: $\frac{1}{3 \cdot 3 + 4} = \frac{1}{13}$ Right limit: $2 \cdot 3 - 1 = 5$ Since left and right limits are not equal, $f$ is not continuous at $x = 3$.

Practice Problems

• Compute limits using algebraic rules and factoring.

• Discuss continuity for various functions, including rational, root, and piecewise functions.

Example:

• Is $f(x) = \frac{x^2 - 25}{x + 5}$ continuous at $x = -5$? For $x \neq -5$, $f(x) = x - 5$. At $x = -5$, $f(-5) = 1$. Limit as $x \to -5$ is $-5 - 5 = -10$. Since $f(-5) \neq \lim_{x \to -5} f(x)$, $f$ is not continuous at $x = -5$.

Additional info: These notes expand on the lecture's brief points, providing definitions, examples, and formulas for limits and continuity, including handling indeterminate forms and piecewise functions.