How one can evaluate $\lim_{t\to0}\frac{f(4t, e^t)}{f(\sin2t, \cos2t)}$?

Question

How one can evaluate $$\lim_{t\to0}\frac{f(4t, e^t)}{f(\sin2t, \cos2t)}$$ where

$f:\mathbb{R^2}\to \mathbb{R}$ be a function with continuous partial derivatives and $f(0,1)=0$ and $f_x(0,1)=1$ and $f_y(0,1)=2$

Thanks in advance.

Answer 1

We have that $f(x,y)$ is differentiable at $(0,1)$ and for $(h,k)\to (0,0)$

$$f(h,1+k)=(1,2)\cdot (h,k)+o(\sqrt{h^2+k^2})$$

therefore

• $f(4t, e^t)=f(4t, 1+t+o(t))=6t+o(t)$
• $f(\sin2t, \cos2t)=f(2t+o(t),1+o(t))=2t+o(t)$