Comparing Functions and Limits (a) If f(2)=4, can you...

Comparing Functions and Limits
(a) If $f(2)=4,$ can you conclude anything about the limit of $f(x)$ as $x$ approaches $2 ?$ Explain your reasoning.
(b) If the limit of $f(x)$ as $x$ approaches 2 is $4,$ can you conclude anything about $f(2) ?$ Explain your reasoning.

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Key Concepts

Continuity

Continuity at a point means that the function’s limit as the input approaches that point equals the actual value of the function at that point. This concept bridges the gap between local behavior (described by limits) and the exact function value, ensuring there are no jumps or discontinuities at that point.

Function Value

The function value at a specific point is the output of the function exactly at that point. It may or may not coincide with the limit as the input approaches that point because the function’s behavior arbitrarily close to the point may differ from the designated output at the point.

Limit

The concept of a limit describes the value that a function approaches as the input approaches a specific point. It reflects the behavior of the function near that point and not necessarily the function's actual value at the point. Understanding limits is fundamental in analyzing how functions behave in a neighborhood around particular points, regardless of what happens exactly at the point.

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