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# Differential Equations/Limits

Question
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

Volunteer

#### Abe Mantell

##### Expertise

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!

##### Experience

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