Calculus Limits, Continuity, Asymptotes, and Derivatives Flashcards

Vocabulary flashcards based on lecture notes covering limits, types of discontinuities, asymptotes, continuity, and derivatives.

$\epsilon - \delta$ Limit Definition

For a limit $\lim_{x \to c} f(x) = L$, for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$.

Removable Discontinuity (Hole)

A type of discontinuity where the limit exists, but the function value at that point is different or missing; occurs when a common factor in the numerator and denominator cancels.

Infinite Discontinuity (Vertical Asymptote)

A type of discontinuity where the function gets infinitely close to an $x$ value but never touches it; occurs when the denominator of the function equals $0$.

Jump Discontinuity

A type of discontinuity where the function approaches different values from the left side and right side.

Horizontal Asymptote

A line where a function gets infinitely closer to a $y$ value but never touches it; found by comparing numerator degree $n$ with denominator degree $m$ for $f(x) = \frac{a x^n}{b x^m}$.

Slant Asymptote

A diagonal line that the function gets closer and closer to as $x$ gets really large or really small; occurs when the degree in the numerator is exactly one more than the degree in the denominator.

Limit Definition of a Derivative

The formal representation of a derivative defined as $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.

Definition of Continuity

A function $f(x)$ is continuous at $c$ if $\lim_{x \to c} f(x) = f(c)$.