Finding Limits Graphically

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Question

Finding Limits Graphically

Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2

a. Find limx→2+ f(x), limx→2− f(x), and f(2).

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the graph of F of X is equal to curly bracket of or minus X X is less than 31 X is equal to 3. And X minus 2X is greater than 3. Determine the limit as X approaches 3 from the right of F of X the limit as X approaches 3 from the left of F of X and F of 3. Awesome. So it appears for this particular prom, we're asked to consider our graph of F of X, which is shown to us by the prom itself. We have a graph that's given to us, and we're ultimately trying to solve for the limit as X approaches 3 from the right of F of X. That's our first answer. Our second answer we're trying to determine is the limit as X approaches 3 from the left of F of X, and our third and final answer is we're trying to determine F of 3. Awesome. So with that said, now that we know we're ultimately trying to solve for three separate answers. Let us first find our, let's focus our attention on determining or finding the limit. As X approaches 3 from the right of F of X. Awesome. So, from the graph, as you can see, as X approaches 3 from the right, that will mean that As a result, F of X will be approaching 1. So once again, when we look at our graph, we could see that as X approaches 3 from the right, F of X will approach 1. So therefore we could safely write that the limit as X approaches 3 from the right of F of X is going to be simply equal to 1. So once again, The limit as X approaches 3 from the right of F of X is going to be simply equal to 1. Now, let's switch gears and find the limit as X approaches 3 from the left of F of X. And as you can see from the graph, as X approaches 3 from the left, that will mean that F of X will approach 1. So, therefore, as a result, the limit as X approaches 3 from the left of F of X will be also equal to 1. Now we can focus on determining F of 3. So we need to note when we are focusing our attention on F of 3, we need to note that there will be 2 key points at. X equals 3. So there's two key points. So let's make a star here and note that we're dealing with two key points. So, with that said, We need to note that there's a solid point at, and let's make a note of this in red, there's a solid point at 3.1, and we also need to note that there's a hollow point or a hole. You might have heard it be called a hole. So there's a solid point once again at, and let's call this one. There's a solid point at 3.1, and then there's a hollow point, let's call this 2, at 3.1. And let us note that the solid point represents the actual value of F of 3, while the hollow point shows a value that the function approaches but does not obtain. So since the solid points determine the function values, we can conclude that once and for all that F of 3 is equal to -1. So therefore, without a doubt, our final three answers are as follows. And we can put a box around our final set of answers. So to quickly recap here, our final answers without a doubt have to be the limit, as X approaches 3 from the right of FF X is equal to 1, and the limit as X approaches 3 from the left of F of X is equal to 1, and F of 3 is going to be equal to -1. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Finding Limits Graphically

Let f(x) = {3 - x , x < 2
2, x = 2
x/2, x > 2

<IMAGE>

a. Find limx→2+ f(x), limx→2− f(x), and f(2).

Key Concepts

Limits

Piecewise Functions

Continuity

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