Differential Equations/Limits

Advertisement


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)

Answer
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

All Answers


Answers by Expert:


Ask Experts

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

©2016 About.com. All rights reserved.