Mathematics 25 Online
OpenStudy (anonymous):

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.

OpenStudy (anonymous):

$\lim_{x,y \rightarrow 0}\frac{x^2}{5x^2+4y^2}$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.

OpenStudy (anonymous):

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

OpenStudy (zarkon):

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

OpenStudy (zarkon):

OpenStudy (anonymous):

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

OpenStudy (anonymous):

lol

OpenStudy (anonymous):

bookmarking for future reference