How can I prove that \(\displaystyle\lim_{{{\left({x},{y}\right)}\to{\left({0},{0}\right)}}}{\frac{{{\left|{x}\right|}^{{{\frac{{{3}}}{{{2}}}}}}{y}^{{{2}}}}}{{{x}^{{{4}}}+{y}^{{{2}}}}}}\rightarrow{0}\)

Lesnaoq73 2022-03-27 Answered
How can I prove that
lim(x,y)(0,0)|x|32y2x4+y20
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Answers (1)

Ronald Martinez
Answered 2022-03-28 Author has 7 answers
I assume you want to show that
lim(x,y)(0,0)|x|32y2x4+y20
Although sometimes it is not the nicest way, switching to polar coordinates is a general approach that will solve these kinds of limits. Let x=rcosθ, y=rsinθ. Then your limit is
lim(x,y)(0,0)|x|32y2x4+y2=limr0r32|cosθ|32r2sin2θr2(r2cos4θ+sin2θ)=limr0r32|cosθ|32(r2cos4θsin2θ+1)
Now, since
0|cosθ|32(r2cos4θsin2θ+1)|cosθ|321
we can apply the squeeze theorem, and since limr0r32=0, we see that the original limit is 0.
Hope that helps.
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