Answer to Question #172760 in Calculus for Yudhvir singh

Question

e) lim_((x y) to (0 0)) (sin x)/(y) exists.

Answer 1

Solution:

$\lim _{\left(x,\:y\right)\to \left(0,\:0\right)}\left(\frac{\sin \left(x\right)}{y}\right)$

We calculate limit along two different paths.

Along $x=0:$

$\lim _{\left(x,\:y\right)\to \left(0,\:0\right)}\left(\frac{\sin \left(0\right)}{y}\right)$$=0$

Along $x=y:$

$\lim _{\left(y,\:y\right)\to \left(0,\:0\right)}\left(\frac{\sin \left(y\right)}{y}\right)$$=1$

Since, they are unequal, limit diverges or does not exist.