SOLUTION: if limit (e ^(ln (a x)) \[Times] tan (2a/(x)))=8 , as x \[LongRightArrow] - \[Infinity] Find the value of a?

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: if limit (e ^(ln (a x)) \[Times] tan (2a/(x)))=8 , as x \[LongRightArrow] - \[Infinity] Find the value of a?      Log On


   

Question 1204233: if limit (e ^(ln (a x)) \[Times] tan (2a/(x)))=8 , as x \[LongRightArrow] - \[Infinity]
Find the value of a?
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The natural log and base 'e' are inverses of each other.
e^(ln(ax)) = ax

As , then ln(ax) approaches positive infinity when a < 0.

The equation

is the same as


The ax portion approaches positive infinity as when a < 0.

The portion approaches tan(0) = 0 as .
This is because the 2a/x part approaches 0 as x approaches positive infinity.

This will mean

turns into

The left hand side is one of the indeterminate forms in calculus.

What you'll need to do is rewrite one of the expressions so that you have a ratio of two functions.

This is one rewrite we could do
ax%2Atan%28%282a%29%2Fx%29 ---> %28tan%28%282a%29%2Fx%29%29%2F%281%2F%28ax%29%29
The new equivalent expression is of the form P/Q where
P+=+tan%28%282a%29%2Fx%29 and Q+=+1%2F%28ax%29

From here, use L'Hopital's rule to apply the derivative to functions P and Q.
Then apply the limit to see if you get another indeterminate form or not.
If so, then apply L'Hopital's rule again.
If not, then you'll be able to solve for 'a'.

I'll let the student take over from here.