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I am really struggling in my Differential Equations class... Thank you so much in advance for your help!! I greatly appreciate it!!!

Does this limit exist...and why or why not??

Limit of: xy/(x^2 + y^2) as (x,y) ---> (0,0)

Hello Laura,

That limit does not exist. If it did, then the limit would be the same from all

directions (i.e. all paths through the origin). If we take a linear path through

(0,0), let y=mx (clearly as x-->0, so does y). Replacing y with mx and taking the

limit as x-->0:

limit (x*mx)/(x^2+(mx)^2)

x-->0

=limit mx^2/((1+m^2)x^2)=m/(1+m^2), which depends on m, the slope.

x-->0

Hence, the limit along various linear paths through the origin gives different

values for different slopes. Thus, the limit does not exist.

OK?

Abe

Differential Equations

Answers by Expert:

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Over 15 years teaching at the college level.**Organizations**

NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.**Education/Credentials**

B.S. in Mathematics from Rensselaer Polytechnic Institute

M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook