In Exercises 39 - 44 , use the graph to estimate the limits...

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist. $$( \mathbf { a } ) \lim _ { x \rightarrow 3 ^ { - } } f ( x )$$ $$( \mathbf { b } ) \lim _ { x \rightarrow 3 ^ { + } } f ( x )$$ $$( \mathbf { c } ) \lim _ { x \rightarrow 3 } f ( x )$$ $$( \mathbf { d } ) f ( 3 )$$

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist.

$$( \mathbf { a } ) \lim _ { x \rightarrow 3 ^ { - } } f ( x )$$

$$( \mathbf { b } ) \lim _ { x \rightarrow 3 ^ { + } } f ( x )$$

$$( \mathbf { c } ) \lim _ { x \rightarrow 3 } f ( x )$$

$$( \mathbf { d } ) f ( 3 )$$

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Key Concepts

Limit

A limit describes the behavior of a function as the input approaches a particular value. It focuses on the value that the function approaches, regardless of whether the function is defined at that point.

One-sided Limits

One-sided limits consider the values of a function as the input approaches the point of interest from one specific direction: either from the left (x ? c?) or from the right (x ? c?). This concept is key when the behavior on either side of a point is different.

Two-sided Limit

The two-sided limit exists if and only if both one-sided limits (from the left and the right) exist and are equal. If they differ, the overall limit at that point does not exist, indicating a possible discontinuity.

Function Value

The function value at a point, denoted as f(c), is the actual output of the function when the input is c. This is distinct from the limit, particularly in cases of removable or jump discontinuities where the function's value may not coincide with its limiting behavior.

Graphical Analysis

Graphical analysis involves using a graph to visually estimate the behavior of a function near a point. By examining how the graph behaves as x approaches a value, one can estimate the one-sided limits, the overall limit, and the actual function value at that point.

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