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. $\lim _{x \rightarrow 25} g(x)$
GRAPH CANT COPY

Key Concepts

Discontinuities

Discontinuities occur when a function is not continuous at a point, even though the limit as the input approaches that point might exist. Different types of discontinuities—such as removable discontinuities (where the limit exists but the function might have a hole) or jump discontinuities (where the limits from either side differ)—are important for understanding how and why a function’s behavior might change abruptly at certain values.

Existence and Non-Existence of Limits

For a limit to exist at a point, the values approached by the function from both sides must be equal. If the left-hand limit and right-hand limit differ, or if the function behaves wildly (for instance, oscillates or becomes unbounded), then the limit at that point does not exist.

Limit of a Function

This concept defines the value that a function approaches as the input gets arbitrarily close to a specific point. It is fundamental in calculus because it provides a way to talk about the behavior of functions near points, regardless of whether the function is defined exactly at that point.

One-Sided Limits

One-sided limits refer to the value a function approaches as its input approaches a particular point from one side only—either from the left or from the right. They are essential for analyzing functions that may behave differently on either side of a point, such as at points of discontinuity.

Transcript

Please give Ace some feedback