Functions, Limits, and Continuity

Vocabulary flashcards covering numbers classification, function concepts, limit evaluation techniques, limit theorems, continuity conditions, discontinuity types, and the difference quotient.

Number Classification Hierarchy

The branching hierarchy categorizing all complex numbers into real and imaginary sets; real numbers divide into rational and irrational numbers; rational numbers branch into integers and non-integers; integers divide into negative integers and whole numbers; whole numbers consist of zero and natural numbers.

Real Function

A relation between two variables $x$ and $y$ such that, for each $x$ in a domain $R$ of real numbers, one or more real values for $y$ are obtained, making $y$ a real function of the real value $x$.

Dependent Variable vs. Independent Variable

In a relation where $y$ depends on $x$, $y$ is designated as the dependent variable (or function) and $x$ is designated as the independent variable.

Function Mapping Criterion

A relation from an input set to an output set is a function if every input maps to exactly one output; it fails to be a function if any single input maps to multiple distinct outputs.

Limit of a Function

A $y$-value that a function tends to generate or approach as the input variable $x$ gets closer and closer to a specified value $a$, denoted as $\lim_{x \to a} f(x) = L$.

Existence of a Limit Theorem

If $f$ is a function and $c$ and $L$ are real numbers, then $\lim_{x \to c} f(x) = L$ if and only if both the left-hand and right-hand limits exist and are equal: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$.

One-Sided Limits

Limits that describe the values approached by $f(x)$ as $x$ approaches $c$ strictly from one direction: the limit from the left is denoted $\lim_{x \to c^-} f(x) = L_1$ and the limit from the right is denoted $\lim_{x \to c^+} f(x) = L_2$.

Indeterminate Form

An expression such as $\frac{0}{0}$ resulting from direct substitution that has no meaning as a real number, indicating that the limit cannot be determined without rewriting the function algebraically.

Dividing Out Technique

An algebraic method for finding limits when direct substitution yields $\frac{0}{0}$, performed by factoring the numerator and denominator, canceling out shared factors, and then evaluating the simplified expression by direct substitution.

Rationalizing Technique

A limit evaluation technique used when direct substitution produces $\frac{0}{0}$ on a fraction containing a radical, involving multiplying both the numerator and the denominator by the conjugate of the radical expression to eliminate the indeterminate form.

Constant Multiple Theorem (Limits)

The theorem stating that the limit of a constant multiple of a function equals the constant multiplied by the limit of that function: $\lim_{x \to a} [k \cdot f(x)] = k \cdot \lim_{x \to a} f(x)$.

Addition and Subtraction Theorem (Limits)

The theorem stating that the limit of a sum or difference of two functions is the sum or difference of their individual limits: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) = L \pm M$.

Multiplication Theorem (Limits)

The theorem stating that the limit of a product of functions equals the product of their individual limits: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = L \cdot M$.

Division Theorem (Limits)

The theorem stating that the limit of a quotient of functions equals the quotient of their individual limits provided the denominator limit is non-zero: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M}$ with $M \neq 0$. If $M = 0$, the limit does not exist.

Power Theorem (Limits)

The theorem stating that the limit of an integer power $p$ of a function is that power applied to the limit of the function: $\lim_{x \to a} [f(x)]^p = (\lim_{x \to a} f(x))^p = L^p$.

Radical / Root Theorem (Limits)

The theorem stating that the limit of the $n$-th root of a function equals the $n$-th root of the limit of the function: $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L}$.

Degree Rules for Limits at Infinity

For rational functions as $x \to \infty$: if the degree of the numerator is greater than the degree of the denominator, the limit is $\infty$; if the degrees are equal, the limit is the ratio of their leading coefficients; if the degree of the numerator is less than the denominator, the limit is $0$.

Continuity at a Point

A function $f(x)$ is continuous at a point $x = a$ if and only if three conditions are satisfied: (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists, and (3) $\lim_{x \to a} f(x) = f(a)$.

One-Sided Continuity

A condition where a function $f$ is continuous from the left at $x = c$ if $f(c) = \lim_{x \to c^-} f(x)$, and continuous from the right at $x = c$ if $f(c) = \lim_{x \to c^+} f(x)$.

Types of Discontinuity

Discontinuities categorized graphically into four types: Removable (a missing point or hole), Jump (left and right limits exist but are unequal), Infinite (values grow without bound toward an asymptote), and Essential (oscillatory or unbounded behavior).

Removable Discontinuity

A point $x = c$ where $\lim_{x \to c} f(x)$ exists, but does not equal $f(c)$ because $f(c)$ is either undefined or assigned a value away from the line of the curve.

Jump Essential Discontinuity

A discontinuity where both the left-hand limit and the right-hand limit exist at $x = c$, but they are unequal to each other ($\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$).

Infinite Essential Discontinuity

A discontinuity occurring when the one-sided limits fail to exist because the function tends toward positive or negative infinity near $x = c$.

Limit of a Difference Quotient

A foundational limit in calculus representing the instantaneous rate of change or derivative of a function, expressed as $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.