Video Transcript
Assume that the limit as 𝑥 approaches negative one of 𝑔 of 𝑥 is equal to three. If 𝑓 of 𝑥 is equal to two to the power of 𝑥, then find the limit as 𝑥 approaches negative one of 𝑓 evaluated at 𝑔 of 𝑥.
In this question, we’re told that the limit as 𝑥 approaches negative one of 𝑔 of 𝑥 is equal to three and we’re also given a function 𝑓 of 𝑥, which is two to the power of 𝑥. We need to use this information to determine the limit as 𝑥 approaches negative one of 𝑓 evaluated at 𝑔 of 𝑥. And we can start by noticing the limit we’re asked to evaluate is the limit of a composition of two functions. So, let’s start by recalling our result about evaluating the limit of the composition of two functions.
We know if we have the limit as 𝑥 approaches 𝑎 of a function 𝑔 of 𝑥 is equal to 𝐿 and we have a function 𝑓 which is continuous at 𝐿, then the limit as 𝑥 approaches 𝑎 of 𝑓 composed of 𝑔 of 𝑥 is equal to 𝑓 evaluated at 𝐿. We can think of this as taking the limit inside of the function or we can also think of this as taking the function outside of the limit. We’ll apply this to our question. We need to know that our function 𝑓 of 𝑥 is an exponential function, which we know is continuous for all real values of 𝑥.
Now to apply this to our question, we know our value of 𝑎 is negative one and our value of 𝐿 is three. Substituting these values into our limit result, we get the limit as 𝑥 approaches negative one of two to the power of 𝑔 of 𝑥 is equal to two cubed, which is just equal to eight. Therefore, we were able to show if the limit as 𝑥 approaches negative one of 𝑔 of 𝑥 is equal to three and 𝑓 of 𝑥 is equal to two to the power of 𝑥, then the limit as 𝑥 approaches negative one of 𝑓 evaluated at 𝑔 of 𝑥 is equal to eight.
