f(x,y)=(sin(x)sin(y))/xy How do you define f at (0,y) in order to ensure that f is continuous everywhere?

Question

Multivariate Calculus

Answer 1

f(x,y) = (sin x)(sin y)/xy
= (sin x)/x (sin y)/y

As x->0, (sin x)/x -> 1 so you need to define f(x,y) at (0,y) as

f(0,y) = (sin y)/y.

Answer 2

f(x,y)=(sin(x)sin(y))/xy well to answere as easily as possible. Since sin(0) = 0 the equation would now look like

f(x,y)=[0*sin(y)]/(0*y), so no matter what y is the answere will always be 0. so,

f(x,y) = 0 at (0,y). Hope this helps you out


P.S. the guy above me is incorrect because when
f(x,y) = (0,y) you cannot simply erase the sin(x)/x because it is equal to zero, it would be represented as
f(x,y)=[0*sin(y)]/(0*y) --> f(x,y) = 0/0 --> f(x,y) = 0

Answer 3

Technically, you can't. You'll have a removal discontinuity at (0,y).

You remove the discontinuity by using a limit as namenot did.

Answer 4

sin(x)/x => 1 as x =>0 so (without proof) I'd try f(0,y)=sin(y)/y.

Answer 5

namenot is not correct. you cannot remove this discontinuity. you can, however remove discontinuity for f(x,y) in (0,0) which is what namenot did. this is a standard example that a function with multiple variables can be continous but it's variable vectors are not necessarily continouos