Answer
Limit does not exist
Work Step by Step
$\lim\limits_{(x,y) \to (0,0)}$ $\frac{xy^2\cos y}{x^2+y^4}$
Evaluating the limit by direct substitution:
$f(0,0)$ = $\frac{0}{0} $
Therefore, we must calculate the limit a different way:
Evaluating limit along the y-axis: $x$ = 0
$f(0,y)$ = $\frac{(0)y^2\cos y}{(0)^2+y^4}$ = $0$
Evaluating limit along the x-axis: $y$ = 0
$f(x,0)$ = $\frac{x(0)^2\cos(0)}{x^2+(0)^4}$ = $0$
Even though both limits are 0, it does not prove that the given limit is 0 so we must evaluate the limit along a different line
Evaluating limit along $x=y^2$:
$f(y^2,y)$ = $\frac{(y^2)y^2\cos y}{(y^2)^2+y^4}$ = $\frac{y^4\cos y}{2y^4}$ = $\frac{\cos y}{2}$
Now,
$\lim\limits_{(x,y) \to (0,0)}$ $\frac{\cos y}{2}$ = $\frac{1}{2}$
Since $\frac{1}{2} \ne 0$, the limit does not exist
