Question

These statements are true or false?Please justify your answer.(i)`lim_(x→0)(1/(x²)-1/(sin²x))` is in `(0/0)` form.(ii)Domain of `f(x,y)=(xy)/(x⁴+y⁴)` is `R²`.

Answer 1

See below.

Explanation:

(i)`lim_(x→0)(1/(x²)-1/(sin²x)) = lim_(x->0)(sin^2x-x^2)/(x^2sin^2x ) = lim_(x->0)((sinx/x)^2-1)/sin^2x` which is of type `(0/0^2)`

(ii)

`f(x,y)=(xy)/(x⁴+y⁴)` The domain of this function is `RR^2-{0,0}`

because

`lim_((x->0),(y->0))f(x,y)` does not exists as a real number