Introduction to Limits (EMATH-111)

Vocabulary flashcards covering core concepts, piecewise functions, limit laws, one-sided/two-sided limit relationships, and infinite limits from EMATH-111.

Signum Function

A function denoted as $\text{sgn}(x)$, defined piecewise as $-1$ if $x < 0$, $0$ if $x = 0$, and $1$ if $x > 0$.

Signum Function Graph

The graphical representation of the signum function $\text{sgn}(x)$, featuring horizontal lines at $y = -1$ for $x < 0$ and $y = 1$ for $x > 0$, with a point at the origin $(0,0)$.

Absolute Value Function

A function denoted as $|x|$, defined piecewise as $-x$ if $x < 0$, $0$ if $x = 0$, and $x$ if $x > 0$.

Absolute Value Function Graph

The V-shaped graph representing the absolute value function $|x|$, symmetric with respect to the y-axis with a vertex at $(0,0)$.

Limit of a Function (Informal Definition)

If all values of the function $f(x)$ approach the real number $L$ as the values of $x$ ($x \neq a$) approach the number $a$, then the limit of $f(x)$ as $x$ approaches $a$ is $L$, expressed as $\lim_{x \to a} f(x) = L$.

Sum Rule for Limits

A limit law stating that if $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, then $\lim_{x \to c} (f(x) + g(x)) = L + M$.

Difference Rule for Limits

A limit law stating that if $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, then $\lim_{x \to c} (f(x) - g(x)) = L - M$.

Constant Multiple Rule for Limits

A limit law stating that if $\lim_{x \to c} f(x) = L$ and $k$ is a real number, then $\lim_{x \to c} (k \cdot f(x)) = k \cdot L$.

Product Rule for Limits

A limit law stating that if $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, then $\lim_{x \to c} (f(x) \cdot g(x)) = L \cdot M$.

Quotient Rule for Limits

A limit law stating that if $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$.

Power Rule for Limits

A limit law stating that if $\lim_{x \to c} f(x) = L$ and $n$ is a positive integer, then $\lim_{x \to c} [f(x)]^n = L^n$.

Root Rule for Limits

A limit law stating that if $\lim_{x \to c} f(x) = L$ and $n$ is a positive integer, then $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L} = L^{1/n}$, assuming $\lim_{x \to c} f(x) = L > 0$ if $n$ is even.

THEOREM 1—Limit Laws

A set of fundamental algebraic rules used to calculate limits of functions, including the Sum Rule, Difference Rule, Constant Multiple Rule, Product Rule, Quotient Rule, Power Rule, and Root Rule.

One-Sided and Two-Sided Limits Relationship

A function $f(x)$ has limit $L$ at $x = a$ if and only if it has both left and right limits there and these one-sided limits are both equal to $L$: $\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$.

Infinite Limits

Limits in which the values of the function $f(x)$ increase or decrease without bound as $x$ approaches a given value $a$, represented as $\lim_{x \to a} f(x) = \infty$.

Limits at Infinity

Limits evaluated as the variable $x$ increases or decreases indefinitely toward positive or negative infinity, represented as $\lim_{x \to \infty} f(x)$.