Simplifying a Tough Multivariable Limit: (8x+8)(2x+3y)^2) / (2x^2 + 16xy - 7y^2)

[trumpet-205]

Homework Statement

Evaluate this limit,

lim (x,y) > (0,0) f(x,y)

where f(x,y) = ((8x+8)(2x+3y)^2) / (sqrt(3x^2 + 14xy + y^2) - sqrt(x^2 -2xy + 8y^2))

Homework Equations

No?

The Attempt at a Solution

I figure the first attempt is to rationalize this fraction. But after I rationalized it, it came out as

f(x,y) = (sqrt(3x^2 + 14xy + y^2) + sqrt(x^2 -2xy + 8y^2)(8x+8)(2x+3y)^2) / (2x^2 + 16xy - 7y^2)

Which to me is way to complicate it, I tried to approach from (x > y) , (y > x) , (x > 0) , (y > 0) but it does not get simplified at all.

 

[jbunniii]

OK, so try working with this part:

(8x+8)(2x+3y)^2) / (2x^2 + 16xy - 7y^2)

Multiply out the numerator so you get one polynomial divided by another, then carry out the division and see what you get.