- #1
kingwinner
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1) lim [x(y^2)] / (x^2 + y^2)
(x,y)->(0,0)
Find the value of the given limit, if it exists.
Using polar coordinates, set x = r cos(theta), y = r sin(theta)
Then, the given limit = lim [r cos(theta) r^2 sin^2(theta)] / r^2
r->0
= lim r [cos(theta) sin^(theta)]
r->0
= 0 since cos(theta) sin^(theta)<=1, i.e. bounded
If I found that the limit is equal to 0 this way, can I conclude immediately that the original limit is 0 too?
What I believe is that for the part using polar coordinates r->0, it seems that it's appraoching the origin through straight lines paths ONLY, however my textbook says that "for the limit to exist, we must get the same result no matter which of the infinite number of paths is chosen"
Thanks for answering!
(x,y)->(0,0)
Find the value of the given limit, if it exists.
Using polar coordinates, set x = r cos(theta), y = r sin(theta)
Then, the given limit = lim [r cos(theta) r^2 sin^2(theta)] / r^2
r->0
= lim r [cos(theta) sin^(theta)]
r->0
= 0 since cos(theta) sin^(theta)<=1, i.e. bounded
If I found that the limit is equal to 0 this way, can I conclude immediately that the original limit is 0 too?
What I believe is that for the part using polar coordinates r->0, it seems that it's appraoching the origin through straight lines paths ONLY, however my textbook says that "for the limit to exist, we must get the same result no matter which of the infinite number of paths is chosen"
Thanks for answering!