$$\lim_{(x,y) \rightarrow (0,0)}[1-\cos(15xy)]/[x^2y^2+x^2y^3]$$

Question

anonymous · Answer

not sure where to go with this one. you should use lhospital right?

yuki · Answer

1-cos(0)/0 =1 this is a famous one

yuki · Answer

lim x->0 sinx / x =1 right ?  you can derive it from there

anonymous · Answer

hrm yeah, i remember that from cal 1 but what where did you get sinx / x?

anonymous · Answer

i see the sinx but

anonymous · Answer

hrm

yuki · Answer

you are missing the point

anonymous · Answer

yeah confused as what to do

yuki · Answer

$$\lim_{z \rightarrow 0} {1-\cos(z) \over z } = 0 $$

yuki · Answer

so if (x,y) -> 0 then xy -> 0 so by this identity you can substitute xy with z

yuki · Answer

does that make sense?

anonymous · Answer

yeah kinda, if each approach 0, then multiplied they would as well i guess?

anonymous · Answer

but wait, does this prove the multivariable limit or do you need to replace x =y x=y^2 all that jazz still?

yuki · Answer

none of that is necessary, as a matter of fact, as long as one of x or y is approaching to 0 the limit you are looking for will be equal to 0

anonymous · Answer

oye, just plugged it into our online homework thing and it said L=0 was wrong

anonymous · Answer

it's strange though, we did these problems last sub-chapter. now we are doing partial derivatives and there isn't a problem like this one in the sub-chapter

anonymous · Answer

i think we are supposed to do a partial derivative or something..?