# Math Help - limits in calculus in several variables

1. ## limits in calculus in several variables

hey,

can't seem to find this limit:

lim((x,y)->(0,0)) ((1+x^2+y^2)^(1/(x^2+y^2+xy^2))

the answer should be "e", but I really don't know how to get there.

appreciate help thanks

2. Originally Posted by mariannehp
hey,

can't seem to find this limit:

lim((x,y)->(0,0)) ((1+x^2+y^2)^(1/(x^2+y^2+xy^2))

the answer should be "e", but I really don't know how to get there.

appreciate help thanks
use polar coordinates to get

$\displaystyle \lim_{r \to 0}[1+r^2]^{\frac{1}{r^2+r^3\cos(\theta)\sin^2(\theta)}}$

Now rewrite as

$\displaystyle \lim_{r \to 0}exp\left(\frac{1}{r^2+r^3\cos(\theta)\sin^2(\the ta)} \ln\left(1+r^2 \right) \right)$

and use L'hospitials rule on the above.

3. We also know that $\displaystyle\lim _{h \to 0} \left( {1 + h} \right)^{\frac{1}{h}} = e$