Calculus Limits and Continuity Vocabulary

Comprehensive vocabulary flashcards covering basic limits, limit laws and algebra, asymptotes, continuity, types of discontinuities, and the Intermediate Value Theorem.

Limit

The value that a function approaches as $x$ approaches a particular number, depending on the behavior of the function near that point rather than its actual value there.

Delta-Epsilon Proof

A formal proof showing that $f(x)$ can be made arbitrarily close to $L$ by making $x$ sufficiently close to $a$.

Informal Meaning of a Limit

The expression \text{\lim}_{x \to a} f(x) = L, meaning that as $x$ gets closer and closer to $a$, $f(x)$ gets closer and closer to $L$.

Direct Substitution

A method of evaluating a limit by plugging the approaching value directly into the function, valid when the function is continuous at that point.

Indeterminate Form

An expression, such as $\frac{0}{0}$, that does not by itself determine the value of a limit.

Algebraic Techniques for $\frac{0}{0}$ Forms

Methods used to rewrite an expression when direct substitution yields an indeterminate form, including factoring and canceling, rationalizing radicals, making substitutions, and simplifying complex expressions.

Factor Cancellation in Limits

The process of canceling a factor that is nonzero near the point being approached to produce an equivalent expression for nearby $x$-values and simplify limit evaluation.

Left-Hand Limit

The value $f(x)$ approaches as $x$ approaches $a$ from values less than $a$, written as \text{\lim}_{x \to a^-} f(x).

Right-Hand Limit

The value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$, written as \text{\lim}_{x \to a^+} f(x).

Two-Sided Limit Existence

The condition where a two-sided limit exists if and only if both the left-hand and right-hand limits exist and are equal (\text{\lim}_{x \to a^-} f(x) = \text{\lim}_{x \to a^+} f(x)).

Squeeze Theorem

A theorem stating that if $g(x) \le f(x) \le h(x)$ near $a$, and both $g(x)$ and $h(x)$ approach the same limit $L$, then $f(x)$ also approaches $L$.

Trigonometric Limits

Limits used to evaluate expressions involving functions such as $\sin(x)$ and $\cos(x)$, especially when direct substitution produces an indeterminate form.

Limit at Infinity

A limit describing the behavior of a function as $x$ becomes very large ($+\infty$) or very negative ($-\infty$).

Vertical Asymptote

A feature that occurs when a function's values grow without bound as $x$ approaches a particular finite value.

Horizontal Asymptote

A value that a function approaches as $x$ approaches $+\infty$ or $-\infty$.

Continuity at $x = a$

The condition fulfilled when $f(a)$ exists, \text{\lim}_{x \to a} f(x) exists, and \text{\lim}_{x \to a} f(x) = f(a).

Removable Discontinuity

A 'hole' in the graph where the limit exists, but the function is either undefined at that point or has a different value there.

Jump Discontinuity

A discontinuity that occurs when the left-hand and right-hand limits both exist but are different.

Infinite Discontinuity

A discontinuity that occurs when a function's values increase or decrease without bound near a particular $x$-value, typically producing a vertical asymptote.

Intermediate Value Theorem (IVT)

A theorem stating that if a function is continuous on an interval, it must take on every value between its values at the endpoints.

Conditions for the Intermediate Value Theorem

The essential requirement that the function must be continuous on the entire closed interval $[a,b]$.