Use the graph of g g g in the figure to find the following values or state that they do not exist. <IMAGE> lim x → 0 g ( x ) {\displaystyle\lim_{x\to0}g\left(x\right)} x → 0 lim g ( x )

Welcome back everyone. In this problem, we want to use the graph of the function K to find the limit of K FX. As X approaches zero here we have our graph below. And for our answer choices, A says the limit is four B five C zero. And D says the limit of K FX as X approaches zero does not exist. Now, how can we figure out the limit of K FX as X approaches zero? What do we know about limits? Well, before we can find the limit, we'll need to establish that the limit exists and recall. OK, that a limit in this case, the limit as X approaches zero of K FX exists. OK. If the right exists, if the limit, OK. As X approaches zero from the left hand side of K FX is equal to the limit as X approaches zero from the right hand side of K FX. In other words, let's evaluate the limit from both sides. And if they are the same, that means the limit exists and it is that value. So let's go to our diagram here is where X equals zero basically along the Y axis. OK. So now we're gonna try and look at our function. OK, look at the value of K or sorry, look at our function K as it approaches as X approaches zero. Let's start from the left. What do you notice as X approaches zero as X approaches zero, K FX approaches four. So the limit, OK. The limit as X approaches zero from the left of K FX equals four. Is that the same as, or is that the same for the limit from the right? Well, yes, for the limit from the right. OK. Notice that as X approaches zero, K also approaches four. OK. So the limit as X approaches zero from the right of K FX is also equal to four. What does that tell us? Well, no, that tells us then that the left hand limit is equal to the right hand limit. That means that the limit exists and the limit then OK, the limit as X approaches zero of K FX is going to be equal to four. Therefore, if we look at our answer choices that tells us a is the correct answer. Thanks a lot for watching everyone. I hope this video helps.

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

2:55 minutes

Problem 2.4b

Textbook Question

Use the graph of $g$ in the figure to find the following values or state that they do not exist. <IMAGE>
${\displaystyle\lim_{x\to0}g\left(x\right)}$

Verified step by step guidance

Identify the behavior of the function g(x) as x approaches 0 from both the left and the right sides on the graph.

Observe the y-values that g(x) approaches as x gets closer to 0 from the left (x -> 0^-).

Observe the y-values that g(x) approaches as x gets closer to 0 from the right (x -> 0^+).

Determine if the left-hand limit and the right-hand limit are equal. If they are, the limit exists and is equal to this common value.

If the left-hand limit and the right-hand limit are not equal, state that the limit does not exist.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's value at points where it may not be explicitly defined. For example, the limit of g(x) as x approaches 0 indicates what value g(x) approaches as x gets closer to 0, which is crucial for analyzing continuity and differentiability.

Continuity

Continuity of a function at a point means that the function is defined at that point, the limit exists, and the limit equals the function's value at that point. If g(x) is continuous at x = 0, then the limit as x approaches 0 will equal g(0). Understanding continuity is essential for determining the behavior of functions and their limits.

Graph Interpretation

Interpreting the graph of a function is crucial for visualizing its behavior, especially when evaluating limits. The graph provides insights into the function's values, trends, and any discontinuities. By analyzing the graph of g, one can determine the limit as x approaches 0 by observing the y-values that g(x) approaches, which aids in answering the question effectively.

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