I am bit confused with subject, suppose we have function: $f(x,y)=\frac{sin(x+xy)}{x}$, and $lim_{(x,y)\to (0,0)}f(x,y)$ is to be found.

Its looks to me very much like $lim_{x\to 0}\frac{sin(x)}{x}=1$, and computationally intuition seems to be right.

How to arrive at this decision, and how to prove that limit indeed exists at $(0,0)$? Latest part bugs me, probably could arrive at $1$ myself, but it makes no point, since I can't prove that point to arrive to does exist.

up vote 0 down vote accepted

We have $\frac{sin(x+xy)}{x}=\frac{sin(x+xy)}{x + xy} * (1 + y)$

To prove $lim_{(x,y)\to (0,0)}\frac{sin(x+xy)}{x + xy}=1 $ just take two arbitrary sequences $x_n, y_n \rightarrow0$. Then $z_n = x_n + x_ny_n \rightarrow 0$ and you can apply the standard limit $lim_{x\to 0}\frac{sin(x)}{x}=1$

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