Calculus I: Limits and Continuity Vocabulary

Comprehensive vocabulary flashcards covering precise definitions, limit rules, continuity types, and theorems for Calculus I exam preparation.

Precise Definition of a Limit

The formal definition stating that $\lim_{x \rightarrow a} f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

Definition of Continuity at a Point

A function $f(x)$ is continuous at a point $a$ if and only if $f(a)$ is defined, $\lim_{x \rightarrow a} f(x)$ exists, and $\lim_{x \rightarrow a} f(x) = f(a)$.

Definition of a Vertical Asymptote

The line $x = a$ is a vertical asymptote of the curve $y = f(x)$ if at least one of the one-sided limits as $x$ approaches $a$ is infinite (that is, $\lim_{x \rightarrow a^-} f(x) = \pm\infty$ or $\lim_{x \rightarrow a^+} f(x) = \pm\infty$).

Definition of a Horizontal Asymptote

The line $y = L$ is a horizontal asymptote of the curve $y = f(x)$ if either $\lim_{x \rightarrow \infty} f(x) = L$ or $\lim_{x \rightarrow -\infty} f(x) = L$.

Squeeze Theorem

If $g(x) \le f(x) \le h(x)$ when $x$ is near $a$ (except possibly at $a$) and $\lim_{x \rightarrow a} g(x) = \lim_{x \rightarrow a} h(x) = L$, then $\lim_{x \rightarrow a} f(x) = L$.

Intermediate Value Theorem (IVT)

Suppose $f(x)$ is continuous on the closed interval $[a, b]$ and let $N$ be any number between $f(a)$ and $f(b)$, where $f(a) \neq f(b)$. Then there exists a number $c$ in $(a, b)$ such that $f(c) = N$.

Special Limit for Sine

The fundamental trigonometric limit formula stating that $\lim_{t \rightarrow 0} \frac{\sin(t)}{t} = 1$.

Removable Discontinuity

A discontinuity at $x = a$ where $\lim_{x \rightarrow a} f(x)$ exists, but either $f(a)$ is undefined or $\lim_{x \rightarrow a} f(x) \neq f(a)$, represented graphically as a hole.

Jump Discontinuity

A discontinuity at $x = a$ where both one-sided limits $\lim_{x \rightarrow a^-} f(x)$ and $\lim_{x \rightarrow a^+} f(x)$ exist and are finite, but are not equal.

Infinite Discontinuity

A discontinuity at $x = a$ where at least one of the one-sided limits $\lim_{x \rightarrow a^-} f(x)$ or $\lim_{x \rightarrow a^+} f(x)$ approaches $\infty$ or $-\infty$.

Continuous from the Left

A function $f(x)$ is continuous from the left at a point $a$ if $\lim_{x \rightarrow a^-} f(x) = f(a)$.

Continuous from the Right

A function $f(x)$ is continuous from the right at a point $a$ if $\lim_{x \rightarrow a^+} f(x) = f(a)$.

Left-Hand Limit

The value that $f(x)$ approaches as $x$ approaches $a$ from values strictly less than $a$, denoted by $\lim_{x \rightarrow a^-} f(x)$.

Right-Hand Limit

The value that $f(x)$ approaches as $x$ approaches $a$ from values strictly greater than $a$, denoted by $\lim_{x \rightarrow a^+} f(x)$.

Indeterminate Form $\frac{0}{0}$

An algebraic expression resulting from direct substitution where both numerator and denominator evaluate to $0$, requiring factoring, rationalization, or algebraic simplification to determine the limit.

Sum of Cubes Formula

The algebraic factoring identity $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, used to simplify algebraic limits containing cubic expressions.

Limit Law for Sums

The property stating that the limit of a sum of two functions equals the sum of their individual limits, provided both limits exist: $\lim_{x \rightarrow a} [f(x) + g(x)] = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x)$.

Limit Law for Products

The property stating that the limit of a product of two functions equals the product of their individual limits, provided both limits exist: $\lim_{x \rightarrow a} [f(x) g(x)] = \left(\lim_{x \rightarrow a} f(x)\right) \left(\lim_{x \rightarrow a} g(x)\right)$.