Use the graph of f in the figure to find the following...

Use the graph of $f$ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. $f(1)$ b. $\lim _{x \rightarrow 1^{-}} f(x) \quad$ 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. $\lim _{n \rightarrow \infty} f(x)$ h. $\lim _{x \rightarrow 3} f(x)$ i. $f(2)$ j. $\lim _{x \rightarrow 2} f(x)$ k. $\lim _{x \rightarrow 2^{+}} f(x)$ l. $\lim _{x \rightarrow 2} f(x)$

Use the graph of $f$ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.
a. $f(1)$
b. $\lim _{x \rightarrow 1^{-}} f(x) \quad$
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. $\lim _{n \rightarrow \infty} f(x)$
h. $\lim _{x \rightarrow 3} f(x)$
i. $f(2)$
j. $\lim _{x \rightarrow 2} f(x)$
k. $\lim _{x \rightarrow 2^{+}} f(x)$
l. $\lim _{x \rightarrow 2} f(x)$

Key Concepts

Graphical Analysis

Graphical analysis involves examining the visual representation of a function to understand its behavior. When evaluating function values and limits, especially one-sided limits, the graph provides essential insights by showing how the function behaves near key points, any jumps or holes, and overall trends.

Function Evaluation

Function evaluation refers to determining the output value of a function at a specific input point. It involves substituting the input value into the function and calculating or reading off the corresponding output. In graphical terms, this is simply finding the y-coordinate of the point where the vertical line through the input value intersects the graph of the function.

Left-Hand Limit

The left-hand limit of a function at a particular point is the value that the function approaches as the input variable approaches the point from values less than the point. This concept is useful for understanding the behavior of functions as they near a point from the left side and is particularly important when the function may not be defined exactly at the point.

Right-Hand Limit

The right-hand limit of a function at a given point is the value the function approaches as the input variable approaches the point from values greater than the point. It helps in analyzing the behavior of a function from the right side of a point, and it is essential to compare it with the left-hand limit when discussing the overall limit at that point.

Two-Sided Limit

A two-sided limit at a point exists only if both the left-hand limit and the right-hand limit exist and are equal. It is a formal way to say that the function approaches a single, unique value from both sides as the input approaches the given point. Discrepancies between the one-sided limits indicate a discontinuity or an undefined two-sided limit.

Types of Discontinuities

Understanding discontinuities, such as removable or jump discontinuities, is crucial when analyzing limits. A removable discontinuity happens when a function is not defined at a point, yet the limit exists as the function approaches that point. In contrast, a jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal, causing the overall limit not to exist.

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