Formal proof that $\lim_{(x,y) \to (0,1)} \frac{x}{\sqrt{y}} =0$

Question

for midterm preparation I am trying to prove that: $$\lim_{(x,y) \to (0,1)} \frac{x}{\sqrt{y}} =0$$ Using the formal $\epsilon - \delta$ definition. I am having trouble with the estimates, Suppose $\delta = \min(1, 1/ \epsilon^2)$ $$\left|\frac{x}{\sqrt{y}} \right| \leq |1/\sqrt{y}| \leq |1/ \sqrt{y-1}| < 1/ \sqrt{\delta} \leq \epsilon $$ Iam doubting about the part where I just bring a factor of $-1$ into the root to obtain a factor of $|y-1|<\delta$. Could anyone point out a standard approach or say if this is valid?

Answer 1

Your argument does not work. Take $\delta =\min \{\frac 1 2, \frac {\epsilon} {\sqrt 2}\}$. Let $|x| <\delta$ and $|y-1| <\delta$. Use the fact that $|y-1| <\delta$ imples $|y-1| <\frac 1 2$ which implies $y >\frac 1 2$. Hence $|\frac x {\sqrt y}| <\frac {|x|} {\sqrt {\frac 1 2}}=\sqrt {2}|x| <\epsilon$ if $|x| <\delta$ and $|y-1| <\delta$.