Question Video: Finding the Integration of a Polynomial Function Using the Power Rule

True or false: If lim_(π‘₯ β†’ 5) 𝑓(π‘₯) = βˆ’3, then 𝑓(5) must be equal to βˆ’3.

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Video Transcript

True or false: If the limit as π‘₯ approaches five of 𝑓 of π‘₯ is equal to negative three, then 𝑓 of five must be equal to negative three.

For this question, we’re given information in the form of a limit. So let us interpret this statement. What this is telling us is that as the value of π‘₯ approaches five, the value of 𝑓 of π‘₯ approaches negative three. Let us now look at the second part of our question. The question is asking us if the limit guarantees that when π‘₯ equals five, the value of the function will be negative three. To answer this, let us recall the general form of a limit.

We recall that the limit concerns values of π‘₯ which are arbitrarily close to π‘Ž but not when π‘₯ is equal to π‘Ž. In our question, this π‘Ž is represented by five. The limit gives us information about our function as π‘₯ approaches five but does not give us information about our function when π‘₯ is equal to five. In fact, 𝑓 of five could be equal to our limit, which is negative three. It could take any other value or it could even be undefined. Since we cannot guarantee that the value of 𝑓 of five is equal to negative three based on the limit, the answer to the question is false. 𝑓 of five does not have to be equal to negative three.

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