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. \begin{equation} \text { a. }\lim _{x \rightarrow 1} g(x) \quad \text { b. } \lim _{x \rightarrow 2} g(x) \quad \text { c. } \lim _{x \rightarrow 3} g(x) \quad \text { d. } \lim _{x \rightarrow 2.5} g(x) \end{equation}

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

Key Concepts

Limit of a Function

This concept involves understanding how a function behaves as the input approaches a particular value. The limit is the anticipated value that the function inches towards, regardless of whether the function is actually defined at that point. It forms the basis of analysis in calculus and is essential for understanding continuity, differentiability, and the overall behavior of functions.

One-Sided Limits

One-sided limits consider the behavior of a function as the input value approaches a target point from only one side—either from the left or from the right. These are crucial in situations where the function behaves differently when approaching from different directions. They help determine whether a limit exists at that point, particularly in cases involving discontinuities or jumps.

Discontinuities

Discontinuities are points at which a function is not continuous, meaning that the function does not have a well-defined limit or the limit does not match the actual function value. This includes jump discontinuities, where the left-hand and right-hand limits exist but differ, and removable discontinuities, where the limit exists even though the function might not be defined at that particular point. Understanding the type of discontinuity is key to evaluating limits correctly.

Graphical Analysis of Limits

Graphical analysis involves examining the graph of a function to understand its behavior near points of interest. By visually inspecting the approach of the function along the x-axis, one can determine the existence and value of limits, identify discontinuities, and distinguish between different types of limit behavior. This technique is especially valuable when dealing with piecewise-defined functions or functions with abrupt changes in behavior.

Transcript

00:01 In this problem, we are given a graph of g of x and we want to find the following limits or explain why they do not exist.

00:08 Our first limit is the limit when x approaches to 1 of g of x.

00:17 For a limit to exist, a limit must exist, must be equal when we approach 1 on the left -hand side and the right -hand side.

00:35 So looking at our graph here, we see that on the left -hand side, the limit when x approaches to 1 is equal to 1.

00:42 But the limit of g of x when x approaches to 1 on the right -hand side is equal to 0.

00:51 Since these two limits do not agree, it means that the limit when x approaches to 1 does not exist.

01:18 Our right does not exist.

01:19 Our d and e for does not exist.

01:22 Our second limit is the limit when x approaches to 2 of g of x.

01:30 Again, let's verify what happens on both the left -hand side and on the right -hand side.

01:41 On the left -hand side, we see that when x approaches to 2, g of x approaches to 1.

01:47 And similarly, on the right -hand side, when x approaches to 2, g of x is equal to 1.

01:52 Since these two limits agree, it means that the limit exists and that limit is equal to 1.

02:22 Next, we have the limit when x approaches to 3 of g of x.

02:38 So as before, let's verify what happens when we approach 3 on both the left and right -hand side.

02:53 So on the left -hand side, the limit is equal to 0.

02:56 On the right -hand side, the limit is also equal to 0.

03:03 Which means that the limit when x approaches to 3 of g of x is equal to 0.

03:09 It exists.

03:10 Although our limit exists, let's note that g of 3 is equal to 1.

03:27 Which means that despite the fact that the limit exists, g of x is actually not continuous at this point.

03:31 We call this a removable discontinuity...

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