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glasskerfu

Answered question

2021-10-22

Calculate the following limits, if they exist, by using a combination of polar coordinates and de L’Hopital rule.

lim_{(x, y) \to (0, 0)} \frac{\arctan (x^{2} + y^{2})}{x^{2} + y^{2}}

l1koV

Skilled2021-10-23Added 100 answers

changing into polar coordinates

x = r \cos θ

y = r \sin θ

x^{2} + y^{2} = r^{2} \cos^{2} θ + r^{2} \sin^{2} θ

= r^{2} (\cos^{2} θ + \sin^{2} θ)

= r^{2} (\cos^{2} θ + \sin^{2} θ = 1)

when

(x, y) \to (0, 0)

r \to 0

Now

lim_{(x, y) \to (0, 0)} \frac{\arctan (x^{2} + y^{2})}{x^{2} + y^{2}}

= lim_{r \to 0} \frac{\arctan (r^{2})}{r^{2}}

= lim_{r \to 0} \frac{\frac{1}{1 + {(r^{2})}^{2}} \cdot 2 r}{2 r}

(applying L'Hopital rule)

= lim_{r \to 0} \frac{1}{1 + r^{4}}

= 1

Jeffrey Jordon

Expert2022-06-26Added 2605 answers

Answer is given below (on video)

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