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

Using the graph representing the function 𝑓(𝑥) = (𝑥 + 3)² + 2, determine lim_(𝑥 → −3) 𝑓(x).

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Video Transcript

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

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

We now want to see the value 𝑓 of 𝑥 approaches as 𝑥 tends to negative three. Let’s start by looking at what happens as our values of 𝑥 approach negative three from the left. As our values of 𝑥 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 𝑥 approach negative three from the right? This means our values of 𝑥 will be bigger than negative three. We can see we get a very similar story. As our values of 𝑥 approach negative three from the right, our outputs are still getting closer and closer to two. So as our values of 𝑥 are getting closer and closer to negative three, our outputs 𝑓 of 𝑥 are getting closer to two.

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