Question

Limits,

http://screencast.com/t/xA9Gd3HUR6

Answer 1

we can interchange y and e and ln e=1 so its same as y^1

Answer 2

they are saying
if ln(y)-> -3 as x-> inf
(the natural log of y approaches -3 when x goes to infinity)

then you can make each side the exponent of e

e^ln(y) --> e^-3
and say that the left side e^ln(y) approaches e^-3 as x goes to infinity

the left side is the same as y, or
y --> e^-3 as x--> infinity

Answer 3

in other words, if you know , for example,

y--> -3

then you can assume that
f(y) --> f(-3) where f() is a function.... as long as the function is "nice" (meaning continuous )

Answer 4

I can understand why this would be correct e^ln(y) --> e^-3

But why this ? ln(y) --> e^-3

Answer 5

We took out e from the left side and they are still equal to each other ?

Answer 6

*** But why this ? ln(y) --> e^-3 ***
It does not say that.

It says ln(y) --> -3

Answer 7

and this ? http://screencast.com/t/A0lu6ifa3t

Answer 8

u see what I mean

Answer 9

http://screencast.com/t/sDx1en6CaatC

Answer 10

Here is what they did:

the problem is find the limit
\[\lim_{x \rightarrow \infty} \left( 1-\frac{ 3 }{ x } \right)^x\]

that looks tricky. So they said... let's take the natural log of that expression and find the limit of this new expression.
\[ \lim_{x \rightarrow \infty} \ln\left( 1-\frac{ 3 }{ x } \right)^x \\
\lim_{x \rightarrow \infty} x\ln\left( 1-\frac{ 3 }{ x } \right) \\
\lim_{x \rightarrow \infty} \frac{\ln\left( 1-\frac{ 3 }{ x } \right)}{\frac{1}{x}}
\]

Answer 11

its a form of limits of 1^infinity which is solved by e^{(f(x)-1)*g(x)} where f(x) tends to 1 and g(x) tends to infinity

Answer 12

they eventually show that the limit is
\[ \lim_{x \rightarrow \infty} \frac{\ln\left( 1-\frac{ 3 }{ x } \right)}{\frac{1}{x}} = -3\]

or
\[ \lim_{x \rightarrow \infty} \ln\left( 1-\frac{ 3 }{ x } \right)^x = -3 \]

next, they undo the natural log by making each side the exponent of e
you get
\[ \lim_{x \rightarrow \infty} \left( 1-\frac{ 3 }{ x } \right)^x= e^{-3} \]

Answer 13

where did "y" go ?

Answer 14

y is
\[ \left( 1-\frac{ 3 }{ x } \right)^x \]

Answer 15

so lny = ln(1 - 3/x) ------> (1 - 3/x)^x = (1 - 3/x)^x ?

Answer 16

lny = xln(1 - 3/x) * * * ** * * * correction

Answer 17

I think they are using y just to simplify the write-up. It is shorter than typing out the original messy expression. You do not need to use y.
You can do it the way I did up above.

Answer 18

thanks for the help

Answer 19

clear as mud ?

Answer 20

Yea ! =)

Answer 21

if you can ask your question a different way maybe there is a way to answer it so it makes sense?