Another limit

Calculus Level 3

Evaluate

$\large \lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^2 -y^2}{x^2+y^2}$

1 ∞ 0 Does not exist -1

1 solution

Brian Charlesworth
Dec 21, 2018

Suppose the origin is approached on the line $y = mx$ for some real value $m$ . Then the given limit becomes

$\displaystyle \lim_{(x,y) \to (0,0)} \dfrac{x^{2}(1 - m^{2})}{x^{2}(1 + m^{2})} = \dfrac{1 - m^{2}}{1 + m^{2}}$ ,

the value of which will depend on $m$ . The limit is therefore not unique, implying that the limit $\boxed{\text{does not exist}}$ .