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Multivariable limit of (x,y)->(1,0) of ln(1+y^2/x^2+xy))

  1. Nov 8, 2012 #1
    multivariable limit of (x,y)--->(1,0) of ln(1+y^2/x^2+xy))

    1. The problem statement, all variables and given/known data
    limit of (x,y)--->(1,0) of ln(1+y^2/x^2+xy))
    Find the limit, if it exists, or show that the limit does
    not exist.


    2. Relevant equations



    3. The attempt at a solution

    so i have: lim(x,y)---->(1,0) ln(1+y^2/x^2+xy)

    i rewrote it as: ln(1+y^2)-ln(x(x+y))
    holding the y constant at 0 i get=lim x-->1 0-ln(x^2)=0
    holding x constant at 1 i get=lim y---> ln(1)-(ln(1)=0

    but my question is,is the limit actually 0? or would i have to approach (1,0) from somwhere else?
    i tried to approach (1,0) from a line (x=1+y) only to end up even more confused

    lim(1+y,y) ln(1+y^2)-ln(1+y(1+2y))=0?

    so does the limit actually = 0? or am i making a mistake somewhere?
     
  2. jcsd
  3. Nov 8, 2012 #2

    Zondrina

    User Avatar
    Homework Helper

    Re: multivariable limit of (x,y)--->(1,0) of ln(1+y^2/x^2+xy))

    Your limit is extremely easy to calculate here. You can simply plug it right in. So yes the limit is actually 0, because ln(1) = 0.
     
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