limit of an absolute value function

Question

how would I go about finding the limit of the following absolute value function as it goes to infinity

$\displaystyle\left|\frac{1}{x} - \frac{1}{y}\right|$

Ive never dealt with multivariable limit functions before?

Answer 1

Claim: $$\lim_{(x,y)\to\infty}\left|\frac{1}{x}-\frac{1}{y}\right|=0$$

Justification: I take $$\lim_{ x \to\infty} \frac{1}{|x|} =0 \tag{1}$$ as known. The double limit $$\lim_{ (x,y) \to\infty} \frac{1}{|x|} =0 \tag{2} $$ follows from (1): if both $x,y$ are large, then $x$ is large. Similarly to (2), $$\lim_{ (x,y) \to\infty} \frac{1}{|y|} =0 \tag{3} $$ Add (2) and (3): $$\lim_{(x,y)\to\infty}\left|\frac{1}{x}\right|+\left|\frac{1}{y}\right|=0$$ The squeeze (based on triangle inequality) finishes the proof: $$0\le \left|\frac{1}{x}-\frac{1}{y}\right|\le \left|\frac{1}{x}\right|+\left|\frac{1}{y}\right|$$