Question

Can anyone help me find
lim (1x^2)/(5x^2 +4y^2) (x,y)->origin
along the line y=mx ? I have already found that along the x-axis it is 1/5, along the y-axis it is 0, and that the limit overall does not exist.

Answer 1

\[\lim_{x,y \rightarrow 0}\frac{x^2}{5x^2+4y^2}\]We have y=mx so inputting this gives:\[\lim_{x \rightarrow 0}\frac{x^2}{5x^2+4(mx)^2}=\lim_{x \rightarrow 0}\frac{1}{5+4m^2}\]Since this doesn't depend on x we get:\[=\frac{1}{5+4m^2}\]So the function has a different limit on every line passing through the origin.

Answer 2

And your evaluation of the limit along the x and y axis is correct. Thus, this function is discontinuous at the origin.

Answer 3

simple way...if x=0 the limit is 0
if y=0 the limit is 1/5

\[0\neq 1/5\] so the limit does not exist

Answer 4

I guess I should read your entire question

since you already noticed this ;)

Answer 5

Yep. But I think he/she wanted to see it evaluated along the lines y=mx as well.

Answer 6

lol

Answer 7

bookmarking for future reference