Limits Graphically - One-Sided Limits

Vocabulary practice flashcards covering graphical limits, one-sided limits, piecewise functions, and conditions for limit existence.

Approach from the Left

The process of looking at the behavior of a function $f(x)$ as $x$ approaches a value $c$ from values strictly smaller than $c$ (looking left), denoted as $\lim_{x \to c^-} f(x)$.

Approach from the Right

The process of looking at the behavior of a function $f(x)$ as $x$ approaches a value $c$ from values strictly larger than $c$ (looking right), denoted as $\lim_{x \to c^+} f(x)$.

Limits When You Can Only Approach from One Side

The graphical rule stating that if there is only one road to $c$, you use that road to evaluate the limit.

Two-Sided Limit Existence

The condition where $\lim_{x \to c} f(x)$ exists if and only if the left-hand limit and the right-hand limit both exist and are equal ($\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$).

Does Not Exist ($\text{DNE}$)

The result assigned to a two-sided limit when the left-hand limit and right-hand limit at $x = c$ do not approach the same value.

Piecewise Function

A function that is defined by different sub-functions across different intervals of the domain (for example, $3 - x$ for $x < 2$ and $\frac{x}{2} + 1$ for $x > 2$).

One-Sided Limits of $f(x) = \begin{cases} 3 - x & x < 2 \\ \frac{x}{2} + 1 & x > 2 \end{cases}$ at $x = 2$

The left-hand limit $\lim_{x \to 2^-} f(x) = 1$ and the right-hand limit $\lim_{x \to 2^+} f(x) = 2$, causing the overall limit $\lim_{x \to 2} f(x)$ to not exist.

Limit Existence on an Open Interval

The property that $\lim_{x \to c} f(x)$ exists at every point $c$ within an open interval, such as $(-1, 1)$ or $(1, 3)$, because the left-hand and right-hand limits agree at each interior point.