Use Figure 1.95 to estimate the following limits, if they...

Use Figure 1.95 to estimate the following limits, if they exist. (a) $\lim _{x \rightarrow 1^{-}} f(x)$ (b) $\quad \lim _{x \rightarrow 1+} f(x)$ (c) $\lim _{x \rightarrow 1} f(x)$ (d) $\lim _{x \rightarrow 2^{-}} f(x)$ (e) $\lim _{x \rightarrow 2^{+}} f(x)$ (f) $\lim _{x \rightarrow 2} f(x)$ (FIGURE CANNOT COPY)

Use Figure 1.95 to estimate the following limits, if they exist.
(a) $\lim _{x \rightarrow 1^{-}} f(x)$
(b) $\quad \lim _{x \rightarrow 1+} f(x)$
(c) $\lim _{x \rightarrow 1} f(x)$
(d) $\lim _{x \rightarrow 2^{-}} f(x)$
(e) $\lim _{x \rightarrow 2^{+}} f(x)$
(f) $\lim _{x \rightarrow 2} f(x)$
(FIGURE CANNOT COPY)

Key Concepts

Limit

A limit is the value that a function approaches as the input approaches a particular point. This concept is fundamental in calculus as it provides a way to understand the behavior of functions at points where they may not be explicitly defined or where they might have discontinuities.

Graphical Analysis of Limits

Graphical analysis of limits involves using the graph of a function to estimate the value of limits by observing the behavior of the function near the point of interest. This visual approach helps identify key characteristics such as trends, breaks, or jumps, and is particularly useful when a function may not be defined at the point itself.

Continuity

Continuity is the property of a function whereby the function does not have any breaks, jumps, or points of discontinuity at a specific point. For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function's value at that point must all be equal. This concept is directly related to the existence of limits at those points.

One-Sided Limit

A one-sided limit considers the value that a function approaches as the input approaches a specific point from one side only, either from the left (denoted as x -> a?) or from the right (denoted as x -> a?). This concept is essential when assessing functions that exhibit different behaviors on either side of a point.

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