For the function g(x) graphed here, find the following...

For the function $g(x)$ graphed here, find the following limits or explain why they do not exist.
a. $\lim _{x \rightarrow 1} g(x)$
b. $\lim _{x \rightarrow 2} g(x)$
c. $\lim _{x \rightarrow 3} g(x)$
d. $\lim _{x \rightarrow 2.5} g(x)$

Key Concepts

Limit

The limit of a function as x approaches a particular value is a fundamental concept in calculus that describes the behavior of the function as its input values get arbitrarily close to that point. It is independent of the actual function value at that specific point and focuses on the trend of the function values around that point.

One-sided Limits

One-sided limits refer to the behavior of the function as the input approaches the point of interest from only one side—either from the left (lim x?a?) or the right (lim x?a?). This concept is crucial when a function behaves differently on either side of a point, affecting whether the two-sided limit exists.

Discontinuities

Discontinuities occur when a function is not continuous at a point. Different types include removable discontinuities, where the limit exists despite the function being undefined at the point, and jump discontinuities, where the left-hand and right-hand limits differ. Recognizing the type of discontinuity is important for determining the existence of limits.

Graphical Analysis of Functions

Graphical analysis involves studying the shape and behavior of a function's graph to understand limits and continuity. By inspecting the graph near the point of interest, one can determine whether the function approaches a specific value (implying a limit) or exhibits behaviors such as jumps or asymptotes that prevent the limit from existing.

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