Delete
Share
This Question is Closed
Dido525
Best Response
You've already chosen the best response.
0
\[\lim_{x \rightarrow \infty}(\frac{ 5x-3 }{ 5x+7 })^{3x+1}\]
anonymous
Best Response
You've already chosen the best response.
0
take the log
take the limit
exponentiate
Dido525
Best Response
You've already chosen the best response.
0
I am allowed to use L'Hospital's rule. I am not sure how I would use it though...
Dido525
Best Response
You've already chosen the best response.
0
So I get it's (infinity/infinity)^(infinity) but I am not sure how to go from there...
Dido525
Best Response
You've already chosen the best response.
0
Yep... I am not sure how that would help though...
anonymous
Best Response
You've already chosen the best response.
0
\[(3x+1)\ln(\frac{5x-3}{5x+7})\]
now you \(\infty\times 0\)
Dido525
Best Response
You've already chosen the best response.
0
Yep... I got there.
anonymous
Best Response
You've already chosen the best response.
0
rewrite as
\[\frac{\ln(\frac{5x-3}{5x+7})}{\frac{1}{3x+1}}\] now you have
\[\frac{0}{0}\] and now you can use l'hopital
Dido525
Best Response
You've already chosen the best response.
0
Hmm Okay... Let's see...
anonymous
Best Response
You've already chosen the best response.
0
probably easier if you write
\[\frac{\ln(5x-3)-\ln(5x+7)}{\frac{1}{3x+1}}\]
Dido525
Best Response
You've already chosen the best response.
0
Okay Thanks!
Dido525
Best Response
You've already chosen the best response.
0
Yep. I know.
anonymous
Best Response
You've already chosen the best response.
0
yw
Dido525
Best Response
You've already chosen the best response.
0
@satellite73 : Okay so I have:
\[\lim_{x \rightarrow \infty} \frac{ \frac{ 5 }{ 5x-3 }-\frac{ 5 }{ 5x+7 } }{ \frac{ -2 }{ (3x+1)^2 } }\]
anonymous
Best Response
You've already chosen the best response.
0
denominator is wrong
anonymous
Best Response
You've already chosen the best response.
0
\[\lim_{x \rightarrow \infty} \frac{ \frac{ 5 }{ 5x-3 }-\frac{ 5 }{ 5x+7 } }{ \frac{ -3 }{ (3x+1)^2 } }\]
Dido525
Best Response
You've already chosen the best response.
0
Chain rule. Right.
anonymous
Best Response
You've already chosen the best response.
0
ugly retricealgebra from here on in
have fun, but you will get a rational expression so limit is same as horizontal asymptote
Dido525
Best Response
You've already chosen the best response.
0
:( . Fair enough....
anonymous
Best Response
You've already chosen the best response.
0
actually it is not that bad
Dido525
Best Response
You've already chosen the best response.
0
Yep. I think I see why.
anonymous
Best Response
You've already chosen the best response.
0
\[-\frac{(3x+1)^2}{3}\times \frac{50}{(5x-3)(5x+7)}\] and not need to multiply out, just look at the ratio of the leading coefficients
anonymous
Best Response
You've already chosen the best response.
0
i mean you have to be a bit careful but the leading coefficient of the numerator is \(9\times 50\) and the leading coefficient of the denominator is \(-3\times 25\) so you the ratio is \(-6\)
anonymous
Best Response
You've already chosen the best response.
0
and don't forget to exponentiate at the end
Dido525
Best Response
You've already chosen the best response.
0
Hmm... Okay.
Dido525
Best Response
You've already chosen the best response.
0
So I got 1/e^6 or e^-6.
anonymous
Best Response
You've already chosen the best response.
0
yup
Dido525
Best Response
You've already chosen the best response.
0
:) . Thanks a million :) .