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Video: Finding the Limit of a Function from Its Graph

Alex Cutbill

If the graph shown represents the function 𝑓(𝑥) = 𝑥 − 3, determine lim_(𝑥 → −1) 𝑓(𝑥).

02:52

Video Transcript

If the graph shown represents the function 𝑓 of 𝑥 equals 𝑥 minus three, determine the limit as 𝑥 tends to negative one of 𝑓 of 𝑥.

We have a graph of the function 𝑓 of 𝑥 equals 𝑥 minus three, which we can use to find this limit graphically. The value of 𝑥 that we’re interested in is negative one, which we highlight on the 𝑥-axis. And we want to know what value 𝑓 of 𝑥 approaches as 𝑥 approaches this value, negative one.

There are two directions in which 𝑥 can approach negative one. 𝑥 can approach negative one from below; that is, with 𝑥 always less than negative one but getting closer and closer to negative one. Or 𝑥 can approach negative one from above; that is, with 𝑥 always being greater than negative one but getting closer and closer to negative one.

We consider these two cases separately, so first the limit as 𝑥 tends to negative one from below of 𝑓 of 𝑥. We pick a value of 𝑥 less than negative one, let’s say negative three, and we see that the point negative three, negative six lies on the graph. And so 𝑓 of negative three is negative six.

Of course we could also have seen this from the definition of the function. But it’s more easy to see from the graph that as we choose other values of 𝑥 getting closer and closer to negative one, but always staying at less than negative one, that 𝑓 of 𝑥 gets closer and closer to negative four. Therefore, the limit as 𝑥 tends to negative one from below of 𝑓 of 𝑥 is negative four.

How about the limit as 𝑥 tends to negative one from above of 𝑓 of 𝑥? It’s the same story. This time we pick a value of 𝑥 is greater than negative one and see what happens when 𝑥 approaches negative one from the other direction; that is, with 𝑥 always being greater than negative one. We see that for this direction, also, the limit is negative four.

For a value of 𝑥 which is greater than negative one but very close to negative one, the value of 𝑓 of 𝑥 is very close to negative four. If the limits from below and above have the same value, and in our case they do, we don’t need to specify whether we’re approaching the limits from below or above.

We can just say that the limit, period, is this value. And so the limit as 𝑥 tends to negative one, period, of 𝑓 of 𝑥 is equal to negative four. Notice that in this case, the limit as 𝑥 tends to negative one of 𝑓 of 𝑥 was just 𝑓 evaluated at 𝑥 equals negative one. This isn’t true for all functions, but it is true for a large number of functions, including all linear functions whose graphs are straight lines.