find the limit of this equation lim(x,y) ->(0,0) x sin (1/y)

i have a similr query.. plz do look.

Multiply top and bottom by 1/y to get lim [(x/y)sin(1/y)]/(1/y) The portion sin(1/y)]/(1/y) goes to one. That leaves you to consider the limit x/y. You try x and y separately; via x, it goes to 0, via y it is undeterminate. You a different result for each, so it suggests, it does not exist. This is an elementary way of finding the limit. There are more vigorous and advanced methods such as setting a sphere around the area.

Let me restate, when you have found that the limit does not exist, this is a very powerful method. More rigorous method may be necessary when the answer suggests the limit exists.

limit is 0, because \[-1\leq\sin{\frac{1}{y}}\leq 1\quad\Rightarrow\quad\lim_{(x,y)\rightarrow(0,0)}x\sin{\frac{1}{y}}=0\cdot[-1,1]=0\]

Good answer. After a second look, my method confirms your answer of 0. The [sin (1/y)]/(1/y) goes to 1 as stated. And the second part is actually the lim of xy. Which if x goes to 0 when y is held constant, and y goes to 0 if x is held constant. Suggests lim is 0. (Once again more rigorous methods can be used because the limit can be approached from infinitely many different directions.)