Video Transcript
True or False: The limit as 𝑥 approaches 𝑎 of 𝑓 evaluated at 𝑥 to the 𝑛th power is equal to the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 all raised to the 𝑛th power. Option (A) true, option (B) false.
In this question, we’re asked to evaluate whether a certain limit result is true or false. And we can immediately notice something interesting about this result. We’re taking the 𝑛th power outside of our limit. So this might remind us of a few results we already know, for example, the power rule for limits or the limit of a composition of functions. However, this is neither of these results because our 𝑛th power is our inner operation and we’re taking it outside of the limit. And since this is not one of our standard limit results, let’s take a look at an example.
Let’s try setting 𝑓 of 𝑥 to be the function one plus 𝑥 and 𝑎 equal to one. Then, we can evaluate the left-hand side of the equation. It’s the limit as 𝑥 approaches one of 𝑓 evaluated at 𝑥 to the 𝑛th power. We substitute 𝑥 to the 𝑛th power into our function 𝑓 of 𝑥. This gives us the limit as 𝑥 approaches one of one plus 𝑥 to the 𝑛th power. And this is an algebraic expression. So we can evaluate this by using direct substitution. We substitute 𝑥 is equal to one. We get one plus one to the 𝑛th power, which is equal to two.
Let’s now evaluate the right-hand side of this equation, the limit as 𝑥 approaches one of 𝑓 of 𝑥 raised to the 𝑛th power. Substituting our expression for 𝑓 of 𝑥, we get the limit as 𝑥 approaches one of one plus 𝑥 all raised to the 𝑛th power. And this time we’re evaluating the limit of a linear function. So we can do this by using direct substitution. We substitute 𝑥 is equal to one, and one plus one is two. So this simplifies to give us two to the 𝑛th power. And the left-hand side of this equation gave us two, and the right-hand side gave us two to the 𝑛th power. And the only way these can be true is if our value of 𝑛 was equal to one.
But then if we did have 𝑛 equal to one, our result would just say the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥, which isn’t a particularly useful result to state. And in either case, the statement claims that this is true for any value of 𝑛, which we’ve shown is false. Therefore, we were able to show it’s false to say the limit as 𝑥 approaches 𝑎 of 𝑓 evaluated at 𝑥 to the 𝑛th power is equal to the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 all raised to the 𝑛th power, which was option (B).
