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) $\lim _{x \rightarrow 0} f(x)$ (b) $\lim _{x \rightarrow 3^{-}} f(x)$ (c) $\lim _{x \rightarrow 3^{+}} f(x)$ (d) $\lim _{x \rightarrow 3} f(x)$ (e) $f(3)$ (See graph)

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) $\lim _{x \rightarrow 0} f(x)$
(b) $\lim _{x \rightarrow 3^{-}} f(x)$
(c) $\lim _{x \rightarrow 3^{+}} f(x)$
(d) $\lim _{x \rightarrow 3} f(x)$
(e) $f(3)$
(See graph)

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Key Concepts

Limit of a Function

The limit of a function as x approaches a certain value describes the behavior of the function near that point, regardless of the function’s actual value at the point. It captures the idea of what value the outputs get arbitrarily close to as the inputs approach a specific value, which is a central notion in calculus.

One-Sided Limits

One-sided limits, such as the left-hand limit and right-hand limit, consider the behavior of the function as the input approaches a target value exclusively from one side (either from below or above). They are crucial for identifying the behavior of functions at points of potential discontinuity, such as jumps or abrupt changes, and determining if an overall limit exists.

Continuity and Discontinuity

Continuity at a point requires that the function's limit as x approaches that point agrees from both sides and equals the actual function value at that point. If the one-sided limits differ or if the function value does not match the common limit, the function is discontinuous at that point. This concept helps in understanding and visualizing points where the graph of a function may have breaks, jumps, or holes.

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