Question

Hi everyone...need a little help!
Lim as x->infinity of [1+(43/x)]^x
I have a picture posted of the whole thing worked out...and the answer is e^43...I just need to discuss the concept of why it is e^43...please take a look at my work and maybe you can discuss with me with a more simplistic example why 43 turns into e^43? Thanks!

Answer 1

is there a pic coming up ? or shall i start ?

Answer 2

sorry...pc wouldn't let me type

Answer 3

one sec...

Answer 4

no problem.
do you know of this formula :
\(\large \lim \limits_{x \rightarrow 0} (1+x)^{1/x}=e\)
?

Answer 5

omg...finally...there it is 18.bmp...sorry...stupid computer...grrrr

Answer 6

never saw that formula before! yikes!

Answer 7

can you possibly work out that formula so I know why that is true?

Answer 8

in your pic, you want to know how 43 turns out to be e^43, right ?
thats because, in your 2nd step, you have written, 'ln'
you took natural log on both sides,
so, L= lim...... (original limit Q)
ln L = lim ln (...) [<----your 2nd step]

and you got ln L =43
which gives L = e^{43}
got this ?

Answer 9

if you want to work out that formula, just take L'Hopitals rule, after taking natural log 'ln' on both sides, in exatly the same way what you did in your question.
you'll easily get it as 'e'

Answer 10

uhmmm...need to process that in my brain for a few moments...:o0

Answer 11

question...you said I took the natural log of both sides of the equation...but I only had an expression which is one sided...a little confused there...

Answer 12

if you have an expression, then you just can't add 'ln' there
\(\large \lim \limits_{x \rightarrow \infty} (1+43/x)^{x} \ne \lim \limits_{x \rightarrow \infty} \ln (1+43/x)^{x} \)

Answer 13

ok got it...but what was the "other side" then? lol

Answer 14

everything looks good except the last line, which should read = 43

then if you make each side a power of e you get the result

Answer 15

if you just want to work with one expression, then use
\( \huge \lim \limits_{x \rightarrow \infty} e^{\ln (1+43/x)^{x}} =e^{ \lim \limits_{x \rightarrow \infty}\ln (1+43/x)^{x} }=e^{43}\)

Answer 16

my both sides meant,
\(let L= \lim \limits_{x \rightarrow \infty} (1+43/x)^{x} \\ \ln L = \lim \limits_{x \rightarrow \infty} \ln (1+43/x)^{x} \)

Answer 17

hmmm...ok let me think for a moment...

Answer 18

\(\huge e^{\ln A}=A\)

Answer 19

haha...ok ok...i think I just had a major breakthrough...let me try and explain and you can tell me if I am correct...

Answer 20

this whole time during the sequence and series chapter, everytime I took the limit of what I thought was an expression, wasn't actually an expression but rather an equation, where the lim of "whatever" actually equalled "l" on the other side of the invisible = sign? lol...is that right? lol

Answer 21

all these limits = "L"?

Answer 22

yes, lim (...anything...) has a value (if its defined)
so you can assume that value = L
but if you want to work it without using 2 sides (with one side only) then also you can..

Answer 23

the first line should show
limit stuff = A
the 2nd line should show

limit ln (stuff) = ln(A)

at the very end you find
ln(A)= 43
and finally
A= e^43

Answer 24

so when I actually get an answer of of a limit problem where I had to take the natural log of one side, I was actually also taking the natural log of "L"...SO TO RID THE "l"SIDE OF THE EQUATION OF THE NATURAL LOG, i HAVE TO RAISE BOTH SIDES AS A POWER OF "E"!? RIGHT?

Answer 25

thats absolutely correct.

Answer 26

LOL...OMG...HARTNN...YOU ARE AWESOME!

Answer 27

and you are awesome too! you understood quite quickly :D

Answer 28

the other way to "fix" this, is replace the 2nd line with

e^ln(stuff)

Answer 29

so then as a rule of thumb, I don't really have to be too concious of that L on the other side unless I take a natural log right? otherwise go about my business? is that safe to say?

Answer 30

yes, i would say yes its safe.

Answer 31

point taken phi...but doing the Lophital rule on e^"scary stuff" could get "scary"...

Answer 32

hey Phi...thanks for all your help also, but hartnn gets the medal this time...thank you both!