Calculate limit by definition of two variables

Question

I am trying to calculate the following limit by definition:

$$ \lim_{ (x,y) \to (1,0)} y*e^x = 0 $$

Answer 1

First, you should remember how to do this in single variable. For example, how would you show

$$ \lim_{x \to 2} 5x = 10 $$

What you want to show is that: For all $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x-2| < \delta$ then $ |5x - 10| < \epsilon $. This can be done as follows:

$$ |5x - 10| = 5|x-2| < 5\delta < \epsilon $$

and so by choosing $\delta = \dfrac{\epsilon}{5}$ we are good.

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Now for your case, it's a two variables so you need to show that: for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $ 0 < \sqrt{(x-1)^2 + (y-0)^2} < \delta$ then $|ye^x - 0| < \epsilon $

Also notice that: $y < \sqrt{(x-1)^2 + (y-0)^2} $

can you take it from here?