Homework Statement
Prove that
\frac{dy}{dx}=\frac{y}{x}
for
\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=10
x is not equal to y which is not equal to 0
The Attempt at a Solution
Tried all the normal methods but none seem to work...anyone have any ideas?
Not sure what you mean. The only restriction I'm guessing would be that any negative roots you get will never be a solution to an exponential expression, where the exponent is not 1.
Let y=3^x
y^2-y-12=0
(y-4)(y+3)=0
y=4\ or\ y=-3
3^x=4\ or\ 3^x=-3
\log 3^x = \log 4
x= \frac{\log 4}{\log3}
x \approx 1.2618595071429
For any value of x, 3^x\neq-3, this solution is extraneous (rejected).
Thanks for the help everyone.
Later, I also found it to work with using this:
\lim_{x \to 4}\left(\frac{2-\sqrt{x}}{3-\sqrt{2x+1}}\right)*\frac{3+\sqrt{2x+1}}{3+\sqrt{2x+1}}
Simplifies to
\lim_{x \to 4}\left(\frac{(2-\sqrt{x})(3+\sqrt{2x+1})}{8-2x}}\right)
to
\lim_{x \to...
Homework Statement
\lim_{x\to 4}\left(\frac{2-\sqrt{x}}{3-\sqrt{2x+1}}\right)
\lim_{x\to 0}\left(\frac{2^2^x {-2^x}}{2^x{-1}}\right)
The Attempt at a Solution
I tried rationalizing but doesn't really help in evaluating these limits.
If a limit cannot be factored, what...