Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How can I prove that $\displaystyle{\lim_{(x, y) \to (0, 0)} \frac{x^3 - y^3}{x^2 + y^2}}=0$? The method I've been taught is the pinching one, where you compare the absolute value of the function to greater limits that are known to equal zero, but I haven't managed to find a comparison that works. Could someone point me in the right direction?

share|improve this question

2 Answers 2

up vote 6 down vote accepted

$$\left|\frac{x^3 - y^3}{x^2 + y^2}\right|\leq \left|\frac{x^3}{x^2 + y^2}\right|+\left|\frac{y^3}{x^2 + y^2}\right|\leq |x| + |y|.$$

share|improve this answer

Hint: If $(x,y)$ is such that both $|x|\le\varepsilon$ and $|y|\le\varepsilon$, then your function $f$ is such that $|f(x,y)|\le2\varepsilon$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.