Refer to the function f shown in the figure. Use the graph...

Refer to the function f shown in the figure. Use the graph to estimate the indicated function values and limits.
(A) $\lim _{x \rightarrow 1^{-}} f(x)$
(B) $\lim _{x \rightarrow 1^{+}} f(x)$
(C) $\lim _{x \rightarrow 1} f(x)$
(D) $f(1)$
(E) Is $f$ continuous at $x=1 ?$ Explain.

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Key Concepts

Continuity

A function is continuous at a point if the limit of the function as the input approaches that point equals the function’s value at that point. This concept ensures that there are no abrupt changes, jumps, or holes in the graph of the function at that point.

Graphical Analysis

Graphical analysis is the process of examining the function’s graph to estimate limits, function values, and continuity. This approach is particularly useful when the function’s expression or precise numerical methods are not readily available, allowing for visual estimation of behaviors such as discontinuities.

Limit

A limit describes the value that a function approaches as the input nears a certain point, regardless of whether the function is defined at that point. It is a key concept in understanding the behavior of functions at points where direct evaluation might be complicated or impossible.

One-Sided Limit

One-sided limits consider the behavior of a function as the input approaches a specific point exclusively from the left side or the right side. They are important when the function has different behaviors on each side of the point, which is critical for identifying jump discontinuities.

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