1
$\begingroup$

Problem: Find the limit of the following functions

a) $\displaystyle \lim_{x \to \infty, \ y\to \infty}$ $\frac{x+y}{x^2 + y^2} $

b) $\displaystyle \lim_{x \to 0,\ y\to 2} \frac{\sin(xy)}{x} $

I know how to make cases when the limit does not exists, but I am having trouble to find the limits when they exists. For example, the limit of the second function is clearly 2, but I couldn't find out how to write it formally (not with the formal definition, but in a way it is correct).

$\endgroup$
2
$\begingroup$

Hints: $$\left|\frac{x+y}{x^2+y^2}\right| \le \frac{|x|}{x^2+y^2} + \frac{|y|}{x^2+y^2} \le \frac1{|x|} + \frac1{|y|}$$ and $$\frac{\sin xy}{x} = y\cdot \frac{\sin xy}{xy}.$$

$\endgroup$
  • $\begingroup$ thanks for your suggestion. In the second case, That was what I was thinking to do. I can use directly that $ lim sinxy / xy $ is the fundamental limit even with 2 variables? Can I use something like " when xy approaches 0" ? Thanks! $\endgroup$ – Giiovanna Oct 15 '13 at 14:00
  • 1
    $\begingroup$ Yes. If $x \to 0$ and $y \to 2$, then $xy \to 0$. $\endgroup$ – njguliyev Oct 15 '13 at 14:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.