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BrighterDays
 one year ago
I have the answer....I just don't understand the steps. Find the limit as x approaches infinity of 1+(a/x)^(bx)  I am putting the original question and the answer from my professor with his explanation in the comments. I just don't understand the steps at all that he has written out.
BrighterDays
 one year ago
I have the answer....I just don't understand the steps. Find the limit as x approaches infinity of 1+(a/x)^(bx)  I am putting the original question and the answer from my professor with his explanation in the comments. I just don't understand the steps at all that he has written out.

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satellite73
 one year ago
Best ResponseYou've already chosen the best response.2is the answer \(e^{ab}\)?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2you can guess it are you supposed to use l'hopital?

BrighterDays
 one year ago
Best ResponseYou've already chosen the best response.0In the comments I put a .png of how he worked it out....I don't understand his substitution with z and how he goes from there....he did say you could use l'hopital but it would be more work???

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2ooh i see the solution now, sorry you are supposed to use algebra, and then the fact that \[e=\lim_{x\to 0}\left(1+x\right)^{\frac{1}{x}}\]

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2the solution you have written is only algebra

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2they say put \(z=\frac{a}{x}\) so the piece inside is now \(1+z\)

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2that part is clear right?

BrighterDays
 one year ago
Best ResponseYou've already chosen the best response.0ugh. OK. so what I don't get is the algebra of: this: if z=a/b how does bx=abz^1?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2the exponential notation may have confused you lets write it without the exponential notation

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2btw it is not \(z=\frac{a}{b}\) but rather \(z=\frac{a}{x}\) right?

BrighterDays
 one year ago
Best ResponseYou've already chosen the best response.0Oh yes, sorry  z=a/x

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2ok step by step \[\large z=\frac{a}{x}\] solve for \(x\) you get \[\large x=\frac{a}{z}\] ok so far?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2multiply both sides by \(b\) you get \[\large bx=\frac{ab}{z}\] right?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2well that is it then \[\left(1+\frac{a}{x}\right)^{bx}\] becomes \[\large (1+z)^{\frac{ab}{z}}\]

BrighterDays
 one year ago
Best ResponseYou've already chosen the best response.0ok  I see that now.

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2or are there still other algebra steps?

BrighterDays
 one year ago
Best ResponseYou've already chosen the best response.0That's it! Thank you. So the fact that e= limx→0 of (1+x)^1/z is just something we need to know, correct? I keep looking for it in my book but can't find it. So, I'll just write it and make a note of it. Thanks again!!

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2well i guess so usually it is written as \[e=\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^x\] but if you change \(x\) to \(\frac{1}{x}\) then you get \[e=\lim_{x\to 0}(1+x)^{\frac{1}{x}}\]

BrighterDays
 one year ago
Best ResponseYou've already chosen the best response.0Great! Thank you so much for all your help!!
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