Answer to Question #107181 in Calculus for Nikesh gautam pandit ji

Question #107181
Find the two repeated limits of the function f(x ,y) = (y-x/y+x) (1+x^2)/1+y^2) at (0,0) . Does
the simultaneous limit of f exist as (x, y) →(0,0) ? Give reasons for your answer.
1
Expert's answer
2020-03-31T13:32:31-0400

"\\lim_{y\\rightarrow0} \\lim_{x\\rightarrow0} \\dfrac {y-x} {y+x}\\cdot \\dfrac {1+x^2}{1+y^2}=\n \\lim_{y\\rightarrow0} 1\\cdot \\dfrac {1}{1+y^2}=1"

"\\lim_{x\\rightarrow0} \\lim_{y\\rightarrow0} \\dfrac {y-x} {y+x}\\cdot \\dfrac {1+x^2}{1+y^2}=\n \\lim_{x\\rightarrow0} -1\\cdot \\dfrac {1+x^2}{1}=-1"

The repeated limits don`t coincide so the simultaneous limit doesn`t exist


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