Question

\[\lim_{(x,y) \to (0,0)} x\ cos\left ({1 \over y} \right )\]

Answer 1

Question for you. What can you say about the size of cos(1/y)?

Answer 2

it goes to 1 as y goes to 0?

Answer 3

In general, what can you say about the function cos(theta), what is its max and min?

Answer 4

absolute value <= 1

Answer 5

yes. Hence what can be bound
\[ \left| x \cos(1/y) \right| \] by?

Answer 6

|x cos(1/y)| <= |x|

Answer 7

Yes. Hence the limit is equal to what?

Answer 8

by squeeze theorem, the limit is 0?

Answer 9

Yes

Answer 10

Awesome, thank you