Video: Discussing the Existence of the Limit of a Piecewise-Defined Function at a Point

Discuss the existence of lim_(π‘₯ β†’ 7) 𝑓(π‘₯) given 𝑓(π‘₯) = {13π‘₯ + 7, if 1 < π‘₯ < 7 and 14π‘₯ + 7, if 7 ≀ π‘₯ < 8.

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Video Transcript

Discuss the existence of the limit as π‘₯ approaches seven of 𝑓 of π‘₯ given 𝑓 of π‘₯ is equal to 13π‘₯ plus seven if one is less than π‘₯ which is less than seven and 14π‘₯ plus seven if seven is less than or equal to π‘₯ which is less than eight.

In this example, our 𝑓 of π‘₯ is a piecewise function. And we are asked to find the limit as π‘₯ approaches seven. Seven is the π‘₯ value at which our function switches between 13π‘₯ plus seven and 14π‘₯ plus seven. In order to find whether our limit exists, we need to check whether the left and right limits exist and if they are equal. We’ll start by considering the left limit. Since π‘₯ is approaching seven from below, we know that π‘₯ must be less than seven. Since π‘₯ is less than seven, we can see from our piecewise definition that 𝑓 of π‘₯ is equal to 13π‘₯ plus seven.

Since this is a polynomial, we can use direct substitution. In order to find this limit, we simply substitute π‘₯ equals seven into 13π‘₯ plus seven. And this gives us that the left limit is equal to 98. Since the limit here is equal to a real constant, we know that this limit must exist. Let’s now consider the limit as π‘₯ approaches seven from above. Since π‘₯ is approaching seven from above, we have that π‘₯ is greater than seven. Therefore, from our piecewise definition, we have that 𝑓 of π‘₯ is equal to 14π‘₯ plus seven which is again a polynomial. And so we can use direct substitution in order to find this limit. Substituting π‘₯ equals seven into 14π‘₯ plus seven, we obtain the limit as π‘₯ approaches seven from the right is equal to 105. Therefore, our limit exists.

Now, we have found that both the left and right limit exist. However, the left limit is equal to 98. And the right limit is equal to 105. Therefore, we can conclude that the limit as π‘₯ approaches seven of 𝑓 of π‘₯ does not exist because the left and the right limit are not equal to each other. In this example, we saw how the limit did not exist because the left and right limits were not equal. This is because there is a jump in the function at the point which we’re trying to take the limit. Therefore, we cannot say that the limit of 𝑓 of π‘₯ approaches a particular point since it depends upon which direction we are approaching the limit as to what the limit could equal.

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