While redoing the tests from this year, I came across this question and its solution. It may just be that I'm forgetting something from a lesson we did back in March, but could you maybe please explain the process of what's going on here and why we're doing it? The part that really gets me is the line in the third column where you switch f(x) to e^ln f(x) and why that works?
Thanks in advance
The original limit is indeterminite form 1^infinity. To help evaluate the limit of f(x) we investigate the limit of lnf(x). This allows us to "bring down" the exponent using a log law. The limit of lnf(x) is then easy to find. We weren't asked to find the limit of lnf(x) though...we were asked to find the limit of f(x). So, we sneakily write f(x) as e^lnf(x). We're using a log law here....a^loga[x]=x (the a is the log base). Remember, ln is just a log with base e. So, the limit of f(x) is the same as the limit of e^lnf(x), but with the latter representation, we can use our knowledge that the limit of lnf(x) equals 1. Make sense?
Ohhhh yes now that makes sense. I wasn't connecting that ln was log base e, and thats why that solution works. Thank you