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

Determine lim_(π₯ β β5) π(π₯) using the graph.

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Video Transcript

Determine the limit of π of π₯ as π₯ approaches negative five using the graph.

From the graph of the function π of π₯, we can see that π of π₯ is a piecewise defined function composed of many subfunctions. We are interested in the behaviour of the function π near the π₯-value negative five. So letβs have a look at the subfunctions near that value. When π₯ is greater than or equal to negative 10 but strictly less than negative five, π of π₯ 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 π₯-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 π₯-value negative five is not included in the domain of this subfunction. When π₯ is greater than or equal to negative five but strictly less than negative three, π of π₯ is defined by the newly indicated straight line subfunction. When π₯ is equal to negative three, π of π₯ 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 weβll stop here for now, as we only need the part of the graph near the π₯-value negative five in order to answer the given question.

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

We see that as π₯ approaches negative five from the left, the sub function approaches the value negative seven, although it doesnβt attain this value as indicated by the hollow circle. Therefore, the limit of π of π₯ as π₯ approaches negative five from the left is negative seven. Now, letβs compute the limit of π of π₯ as π₯ approaches negative five from the right. In order to do this, we will consider values of π₯ near negative five but strictly greater than negative five. Therefore, we need to consider the subfunction of π whose domain is values of π₯ greater than or equal to negative five but strictly less than negative three.

We see that as π₯ 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 π of π₯ as π₯ approaches negative five from the right is equal to four. Since four is not equal to negative seven, we have that the limit of π of π₯ as π₯ approaches negative five from the left is not equal to the limit of π of π₯ as π₯ approaches negative five from the right. Therefore, the limit of π of π₯ as π₯ approaches negative five does not exist. This is our final answer.