Finding limits from a graph Use the graph of f in the figure...

Finding limits from a graph Use the graph of $f$ in the figure to find the following values, if they exist. If a limit does not exist, explain why. a. $f(1)$ b. $\lim _{x \rightarrow 1^{-}} f(x)$ c. $\lim _{x \rightarrow 1^{+}} f(x)$ d. $\lim _{x \rightarrow 1} f(x)$ e. $f(3)$ f. $\lim _{x \rightarrow 3^{-}} f(x)$ g. $\lim _{x \rightarrow 3^{+}} f(x)$ h. $\lim _{x \rightarrow 3} f(x)$ i. $f(2)$ j. $\lim _{x \rightarrow 2^{-}} f(x)$ $\mathbf{k}, \lim _{x \rightarrow 2^{+}} f(x)$ 1. $\lim _{x \rightarrow 2} f(x)$ (GRAPH CAN'T COPY)

Finding limits from a graph Use the graph of $f$ in the figure to find the following values, if they exist. If a limit does not exist, explain why.
a. $f(1)$
b. $\lim _{x \rightarrow 1^{-}} f(x)$
c. $\lim _{x \rightarrow 1^{+}} f(x)$
d. $\lim _{x \rightarrow 1} f(x)$
e. $f(3)$
f. $\lim _{x \rightarrow 3^{-}} f(x)$
g. $\lim _{x \rightarrow 3^{+}} f(x)$
h. $\lim _{x \rightarrow 3} f(x)$
i. $f(2)$
j. $\lim _{x \rightarrow 2^{-}} f(x)$
$\mathbf{k}, \lim _{x \rightarrow 2^{+}} f(x)$
1. $\lim _{x \rightarrow 2} f(x)$
(GRAPH CAN'T COPY)

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Key Concepts

Graphical Analysis of Limits

Analyzing limits from a graph involves observing how the function behaves as it gets arbitrarily close to a point, including interpreting open or closed points (indicating whether the point is included in the function) and discontinuities. This analysis can reveal jump discontinuities or removable discontinuities where the limits exist even if the function value does not match the limit.

Two-Sided Limit

The two-sided limit of a function as x approaches a particular value exists only if both the left-hand limit and the right-hand limit exist and are equal. It describes the behavior of the function from both sides of that point, irrespective of the function’s actual value at that point.

Function Value

The function value at a specific input is the output that the function assigns to that input. It represents the actual value of the function at a given point and is determined by direct evaluation or by noting the plotted point on a graph.

One-Sided Limit

A one-sided limit refers to the value that a function approaches as the variable approaches a specific point from one side—either from the left (denoted as the limit from the left) or from the right (denoted as the limit from the right). It is determined by looking at the behavior of the function exclusively on one side of the target point.

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