How to take the limit

[fenixtx423]

Is there any way to solve for this limit? Thanks so much in advance

Limit as X -->∞ of (X/(X+Y))^(-XZ)

 

[HallsofIvy]

Is there any way to solve for this limit? Thanks so much in advance

Limit as X -->∞ of (X/(X+Y))^(-XZ)

Are we to assume that Y and Z are independent of X so can be treated as constants?

Since x occurs both in base and exponent, I would try taking the logarithm.

If \(\displaystyle U= \frac{X}{(X+ Y)^{-XZ}}= X(X+Y)^{XZ}\) then \(\displaystyle ln(U)= XZ log(X+ Y)+ log(X)\)

 

[pka]

Is there any way to solve for this limit? Thanks so much in advance
Limit as X -->∞ of (X/(X+Y))^(-XZ)

Here is a different take on it.
\(\displaystyle \Large\displaystyle{\lim _{x \to \infty }}{\left( {\frac{x}{{x + y}}} \right)^{ - xz}} = {\lim _{x \to \infty }}{\left( {1 + \frac{{ - y}}{{x + y}}} \right)^{ - zx}} = {e^{yz}}\)

 

[fenixtx423]

Wow thanks that is great!