For the function f whose graph is given, state the value of...

For the function f whose graph is given, state the value of the given quantity if it exists. If it does not exist, explain why. $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 0} f(x)} & {\text { (b) } \lim _{x \rightarrow 3} f(x) \quad \text { (c) } \lim _{x \rightarrow 3^{-}} f(x)} \\ {\text { (d) } \lim _{x \rightarrow 3} f(x)} & {\text { (e) } f(3)}\end{array} $$

For the function f whose graph is given, state the value of the given quantity if it exists. If it does not exist, explain why.
$$
\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 0} f(x)} & {\text { (b) } \lim _{x \rightarrow 3} f(x) \quad \text { (c) } \lim _{x \rightarrow 3^{-}} f(x)} \\ {\text { (d) } \lim _{x \rightarrow 3} f(x)} & {\text { (e) } f(3)}\end{array}
$$

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Key Concepts

Function Value versus Limit

This concept distinguishes the actual value of the function at a point from the limit as the variable approaches that point. The function value may not coincide with the limit, especially at points of discontinuity, highlighting the necessity to analyze limits and function values separately.

Existence of a Limit

A limit exists at a point if and only if the left-hand limit and the right-hand limit are both defined and equal. If these one-sided limits differ, the overall limit at that point does not exist, emphasizing the importance of checking continuity from both sides.

Limit of a Function

This concept refers to the value that a function approaches as the input variable gets arbitrarily close to a specific point. It focuses on the behavior of the function near that point rather than its actual value at the point, which is fundamental in calculus.

One-Sided Limits

These limits consider the value that a function approaches from one direction only—either from the left or the right of a given point. They are important in analyzing functions with different behaviors on different sides of a point or when the function is defined piecewise.

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