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

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

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

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Key Concepts

Function Evaluation

Evaluating a function at a specific point involves determining its actual value at that input, which may or may not match the limit as the input approaches that point. Discrepancies between the limit and the function's value can indicate discontinuities such as holes or jump discontinuities.

Limit Existence Criterion

The existence of a limit at a point requires that the left-hand limit and right-hand limit are both defined and equal. If the one-sided limits differ or if one of them does not exist, then the overall limit at that point does not exist.

Limit

A limit describes the value that a function approaches as its input nears a particular point. It does not necessarily depend on the function’s value at that point, but rather on the behavior of the function as the input gets arbitrarily close to that point.

One-Sided Limits

One-sided limits evaluate the behavior of a function as the input approaches a specific point from one direction only—either from the left (denoted by a superscript -) or from the right (denoted by a superscript +). They are crucial for understanding the function’s behavior near boundaries or discontinuities.

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