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Does this limit exist lim e^(r^2)/(cos(t)sin(t)) for (r,t)->(0,0)

  1. Apr 12, 2005 #1
    In my exam the question was to determine the existence of this limit
    [tex] \lim_{(r,t)\rightarrow(0,0)} \frac{e^{r^2}}{\cos{t}\sin{t}} [/tex]

    now i wrote the numerator has no t associated with it so grows or shrinks without bounds, the same applies for the denominator...
    so the limit does not exist

    is this a good reason

    another question was
    [tex] \lim_{(x,y)\rightarrow(0,0)} \frac{\sin{xy}}{xy} [/tex]
    i substituted xy = u and got [tex] \lim_{u \rightarrow 0} \frac{\sin{u}}{u} = 1 [/tex]


    is this the correct method?
    Am i right?
     
  2. jcsd
  3. Apr 12, 2005 #2

    HallsofIvy

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    Staff Emeritus
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    The numerator does not "grow or shrink without bound". As r goes to 0, the numerator goes to 1.

    However, you are correct that the denominator goes to 0 as t does no matter what r is. Since the numerator goes to 1, what does that tell you about the fraction?

    I was a bit suspicious about your second method, but yes, it works, since u is a variable, you are not assuming any relationship between x and y.
     
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