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. $f(1) \quad$ b. $\lim _{T \rightarrow 1^{-}} f(x) \quad$ c. $\lim _{t \rightarrow 1^{+}} f(x)$ d. $\lim _{x \rightarrow 1} f(x)$ (GRAPH CAN'T COPY)

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. $f(1) \quad$ b. $\lim _{T \rightarrow 1^{-}} f(x) \quad$ c. $\lim _{t \rightarrow 1^{+}} f(x)$
d. $\lim _{x \rightarrow 1} f(x)$
(GRAPH CAN'T COPY)

Key Concepts

Graphical Analysis of Functions

Graphical analysis involves interpreting the graph of a function to determine various properties such as function values, limits, and continuity. Reading key features like intersections, open and closed dots, and abrupt changes allows one to extract important information about the behavior of the function at specific points and intervals.

Discontinuity (Jump Discontinuity)

A jump discontinuity occurs when the left-hand limit and the right-hand limit of a function at a point exist but are not equal, causing the overall limit at that point to not exist. This type of discontinuity is important to identify because it indicates an abrupt change in the function's value, which may affect the function's continuity and integrability.

Function Value

A function value, denoted as f(a), is the output of the function when the input is exactly a. This is determined by directly substituting the value into the function, or in the case of a graph, by reading the y-coordinate at x = a. It reflects the value of the function at that specific point and is separate from any limits approaching that point.

One-Sided Limits

One-sided limits describe the behavior of a function as the input approaches a given value from one side only – either from the left (denoted as the left-hand limit) or from the right (the right-hand limit). These limits are critical in understanding the behavior of functions near points of potential discontinuity, especially when the behavior is different from either side.

Overall Limit

The overall or two-sided limit of a function as x approaches a point exists only if both the left-hand and right-hand limits exist and are equal. This concept is fundamental in calculus as it describes the expected behavior of the function at a point, even if the function value at that point is not defined or is different from the limit.

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