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

Video Transcript

Determine the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five using the graph.

From the graph of the function 𝑓 of 𝑥, we can see that 𝑓 of 𝑥 is a piecewise defined function composed of many subfunctions. We are interested in the behaviour of the function 𝑓 near the 𝑥-value negative five. So let’s have a look at the subfunctions near that value. When 𝑥 is greater than or equal to negative 10 but strictly less than negative five, 𝑓 of 𝑥 is defined by the indicated straight line subfunction. Note that the filled-in green circle at the left endpoint of this line indicates that the 𝑥-value negative 10 is included in the domain of this subfunction.

The hollow green circle at the right endpoint of this line indicates that the 𝑥-value negative five is not included in the domain of this subfunction. When 𝑥 is greater than or equal to negative five but strictly less than negative three, 𝑓 of 𝑥 is defined by the newly indicated straight line subfunction. When 𝑥 is equal to negative three, 𝑓 of 𝑥 is defined by the filled-in green circle at the coordinates negative three, negative two. We could carry on doing a similar analysis for the rest of the graph. But we’ll stop here for now, as we only need the part of the graph near the 𝑥-value negative five in order to answer the given question.

We know that if the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five exists, then it must be equal to the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the left and the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the right. Let’s compute the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the left. In order to do this, we will consider values of 𝑥 near negative five but strictly less than negative five. Therefore, we need to consider the subfunction of 𝑓 whose domain is values of 𝑥 greater than or equal to negative 10 but strictly less than negative five.

We see that as 𝑥 approaches negative five from the left, the sub function approaches the value negative seven, although it doesn’t attain this value as indicated by the hollow circle. Therefore, the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the left is negative seven. Now, let’s compute the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the right. In order to do this, we will consider values of 𝑥 near negative five but strictly greater than negative five. Therefore, we need to consider the subfunction of 𝑓 whose domain is values of 𝑥 greater than or equal to negative five but strictly less than negative three.

We see that as 𝑥 approaches negative five from the right, the subfunction approaches the value four and actually attains this value as indicated by the filled-in green circle. Therefore, the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the right is equal to four. Since four is not equal to negative seven, we have that the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the left is not equal to the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five from the right. Therefore, the limit of 𝑓 of 𝑥 as 𝑥 approaches negative five does not exist. This is our final answer.