If $\lim_{x \to \infty} (f(x)-g(x)) = 0$ then $\lim_{x \to \infty} (f^2(x)-g^2(x)) = 0\ $: False?

Question

If $\lim_{x \to \infty} (f(x)-g(x)) = 0$, then $\lim_{x \to \infty} (f^2(x)-g^2(x)) = 0$

My teacher said that the above is false, but I can't find any example that shows that it's false! Can someone explain to me why it's false and also give an example?

You can show that if $f(x)$ is bounded, then the second limit is zero, too. — Thomas Andrews, Nov 12 '15 at 13:22

score 10 · Answer 1 · answered Nov 12 '15 at 13:08

10

Take $f(x)=x$ and $g(x)=x+\dfrac{1}{x}$

answered Nov 12 '15 at 13:08

RFZ

16,814

score 1 · Answer 2 · answered Nov 12 '15 at 15:31

1

Another example: Let $f(x)=\sqrt{x+1}, g(x)=\sqrt{x}$.

answered Nov 12 '15 at 15:31

xpaul

44,000

If $\lim_{x \to \infty} (f(x)-g(x)) = 0$ then $\lim_{x \to \infty} (f^2(x)-g^2(x)) = 0\ $: False?

2 Answers2