How to solve this limit problem?

Question

Find the limit and prove that exists lim (x,y) approach (0,0) (sqrt(x^2y^2+1) - 1) / (x^2 + y^2)

Please explain how to do this problem.

kb · Answer

lim((x,y)→(0,0)) [√(x^2y^2 + 1) - 1] / (x^2 + y^2)
= lim((x,y)→(0,0)) [√(x^2y^2 + 1) - 1][√(x^2y^2 + 1) + 1] 
/ {(x^2 + y^2) [√(x^2y^2 + 1) + 1]}, via conjugates

= lim((x,y)→(0,0)) x^2y^2 / {(x^2 + y^2) [√(x^2y^2 + 1) + 1]}
= lim((x,y)→(0,0)) 1/(√(x^2y^2 + 1) + 1) * lim((x,y)→(0,0)) x^2y^2/(x^2 + y^2)

= (1/2) * lim((x,y)→(0,0)) x^2y^2/(x^2 + y^2) 
= (1/2) * lim(r→0+) (r cos θ)^2 (r sin θ)^2 / r^2, via polar coordinates
= (1/2) * lim(r→0+) r^2 (cos θ sin θ)^2 
= (1/2) * 0
= 0.
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For the next to last step, we use the Squeeze Theorem, noting that
0 ≤ r^2 (cos θ sin θ)^2 ≤ r^2, and lim(r→0+) 0 = 0 = lim(r→0+) r^2.
So, lim(r→0+) r^2 (cos θ sin θ)^2 = 0, as required.
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I hope this helps!

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How to solve this limit problem?

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