Calculus: Definitions of Limits (Section 2.2)

Vocabulary flashcards defining two-sided limits, one-sided limits, conditions for existence and non-existence, and methods for evaluating limits from Section 2.2.

Two-Sided Limit ($\lim_{x \to a} f(x) = L$)

Let $f(x)$ be defined when $x$ is near the number $a$, except possibly at $a$. If the values of $f(x)$ get closer and closer to the number $L$ as $x$ gets closer and closer to $a$ (from either side, with $x \neq a$), then the limit of $f(x)$ as $x$ approaches $a$ is $L$. Alternatively written as $f(x) \rightarrow L$ as $x \rightarrow a$.

Left-Hand Limit ($\lim_{x \to a^-} f(x) = L$)

The limit of $f$ as $x$ approaches $a$ from the left equals $L$, meaning that the value of $f(x)$ gets closer and closer to $L$ as $x$ gets closer and closer to $a$ with $x < a$.

Right-Hand Limit ($\lim_{x \to a^+} f(x) = L$)

The limit of $f$ as $x$ approaches $a$ from the right equals $L$, meaning that the value of $f(x)$ gets closer and closer to $L$ as $x$ gets closer and closer to $a$ with $x > a$.

Condition for Two-Sided Limit Existence

If both one-sided limits exist and are equal to the same number $L$ ($\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$), then the two-sided limit exists and is also equal to $L$ ($\lim_{x \to a} f(x) = L$).

Non-Existence of Limit: Unequal One-Sided Limits

If both one-sided limits exist but have different values ($\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$), then the two-sided limit does not exist ($\lim_{x \to a} f(x) \text{ D.N.E.}$).

Non-Existence of Limit: Non-Existent One-Sided Limit(s)

If either or both one-sided limits do not exist because $f(x)$ does not get close to some number $L$ from one or both sides ($\lim_{x \to a^-} f(x) \neq L$ and/or $\lim_{x \to a^+} f(x) \neq L$), then the two-sided limit does not exist ($\lim_{x \to a} f(x) \text{ D.N.E.}$).

Behavior Near $a$ vs. At $a$ Principle

The principle that when finding $\lim_{x \to a} f(x)$, the limit exists independently of whether or not $f$ is defined at $a$; all that matters is how $f$ is defined near $a$.

Unbounded Behavior (Cause of Limit D.N.E.)

A condition where a limit fails to exist because $f(x)$ does not approach any finite number, but instead increases or decreases without bound from the left and/or right side of $a$.

Oscillating Behavior (Cause of Limit D.N.E.)

A condition where a limit fails to exist because the function oscillates indefinitely between different values (such as $-L$ and $L$) rather than settling toward a single number as $x$ approaches $a$.

Graphical Method (for Determining Limits)

A method of estimating or determining limits by inspecting the behavior of a function's curve on a graph; noted as not always reliable.

Numerical (Tabular) Method (for Determining Limits)

A method of estimating limits by computing values of $f(x)$ in a table as $x$ takes values closer and closer to $a$; noted as not always reliable.

Algebraic (Analytical) Method (for Determining Limits)

A method of determining limits using mathematical formulas and properties (covered in Section 2.3); noted as always reliable when applicable.