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Question Video: Finding the Limit of a Function from Its Graph Mathematics • 12th Grade

Using the graph representing the function 𝑓(π‘₯) = (π‘₯ + 3)Β² + 2, determine lim_(π‘₯ β†’ βˆ’3) 𝑓(x).

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Video Transcript

Using the graph representing the function 𝑓 of π‘₯ is equal to π‘₯ plus three all squared plus two, determine the limit as π‘₯ approaches negative three of 𝑓 of π‘₯.

We’re given a graph, and we’re told that the curve in this graph represents the function 𝑓 of π‘₯ is equal to π‘₯ plus three all squared plus two. We need to use this graph to determine the value of the limit as π‘₯ approaches negative three of the function 𝑓 of π‘₯. First, let’s recall what we mean by the limit as π‘₯ approaches negative three of the function 𝑓 of π‘₯. This is the value that our outputs of 𝑓 of π‘₯ approach as π‘₯ tends to negative three. In other words, we want to see what happens to the outputs of 𝑓 of π‘₯ as our inputs π‘₯ get closer and closer to negative three. So let’s mark negative three on our π‘₯-axis. Remember, the outputs of our function will be their 𝑦-coordinate.

We now want to see the value 𝑓 of π‘₯ approaches as π‘₯ tends to negative three. Let’s start by looking at what happens as our values of π‘₯ approach negative three from the left. As our values of π‘₯ approach negative three from the left, we can see something interesting. By remembering that our outputs are represented by the 𝑦-coordinate, we can see this is approaching two. But we can then ask the question, what happens as our values of π‘₯ approach negative three from the right? This means our values of π‘₯ will be bigger than negative three. We can see we get a very similar story. As our values of π‘₯ approach negative three from the right, our outputs are still getting closer and closer to two. So as our values of π‘₯ are getting closer and closer to negative three, our outputs 𝑓 of π‘₯ are getting closer to two.

Therefore, by using the graph representing the function 𝑓 of π‘₯ is equal to π‘₯ plus three all squared plus two, we were able to show the limit as π‘₯ approaches negative three of 𝑓 of π‘₯ is equal to two.

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