Use the graph of f in the figure to evaluate the function or a...

Question

Use the graph of f in the figure to evaluate the function or analyze the limit.

lim x→3 f(x)

Accepted Answer

Hi, everyone, let's take a look at this practice problem dealing with limits. This problem says use the following graph to evaluate the limit as X approaches -4 of F of X. Below the question we're given a graph that shows F of X, and this is a piecewise function. The first piece is parabolic and begins around the point of -7, 10 and goes to the point of -4.2. The next segment is linear and begins at the point of -4.4 and goes to the point of -2.0, and that initial point of -4.4 is excluded from our function. The next segment is parabolic and it's opening upwards, and it begins at the point of -2.0 and ends at the point of 2.0, and it has a minimum located at 0.3. However, this point is excluded from our function. We have an individual point for our value at X equal to 0, and it is located at 0.1. The last segment is linear and begins at the point of 2.1, and ends around the point of 10. minus 5. And our initial point of 2.1 is actually excluded from our function. We're also given 4 possible choices as our answers. For choice A, we have 0, for choice B, we have -2. For choice C, we have -4, and for choice D, we have the limit does not exist. So we need to determine the limit as X approaches -4 for our function. So in order to do that, we need to first determine if the limit actually exists. And for a limit to exist, if you recall, the limit must be the same whether I'm approaching that value for X from the left hand side or the right hand side of that point. So we're actually going to calculate two limits, one from the left hand side and one from the right hand side. So we're going to start off by looking at our left hand limit. And so here we're gonna be looking at the limit. As X approaches. -4 from the left of F of X. And so here we're going to begin to the left of our point of -4 and follow the function to see what value it actually is approaching. And so doing so, we can see that our function is approaching the value of -2. So this limit is going to be equal to -2. We'll do something similar for the right hand limit. So for the right hand limit, for me looking at the limit. As X approaches minus 4 from the right of F of X. So here we're going to be looking to the right of the point X is equal to -4, and we're going to follow our function as it is approaching the value of X is equal to -4. In doing so, we see that our function is approaching the value of -4. That means that this limit is going to be equal to -4. Now notice that these two limits are actually different. They have different values. And so in order for the limit to actually exist as X approaches minus 4, these two values would need to be equal. So that means that our limit does not exist, so we'll write that as DNE. Which represents does not exist. And so that is actually the answer to this question. And so if we come down here to our answer choices, this actually corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>
lim x→3 f(x)

Key Concepts

Limits

Continuity

Graphical Analysis

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