Calculate the following limits, if they exist, by using a combination of polar c

Question

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{1 - \cos (x^{2} + y^{2})}{x^{2} + y^{2}}

Answer 1

Given: The function is $lim_{(x, y) \to (0, 0)} \frac{1 - \cos (x^{2} + y^{2})}{x^{2} + y^{2}}$
Consider $x = r \cos θ$ , and $y = r \sin θ$ .
The expression becomes $lim_{r \to 0} \frac{1 - \cos (r^{2})}{r^{2}}$
Conclusion:
Apply L’Hopital rule as follows:
$lim_{r \to 0} \frac{1 - \cos (r^{2})}{r^{2}} = lim_{r \to 0} \frac{- (- \sin (r^{2}) \cdot 2 r)}{2 r}$
$= lim_{r \to 0} \sin (r^{2})$
$= \sin (0)$
$= 0$
Hence, $lim_{(x, y) \to (0, 0)} \frac{1 - \cos (x^{2} + y^{2})}{x^{2} + y^{2}} = 0$