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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?

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  • $\begingroup$ You can show that if $f(x)$ is bounded, then the second limit is zero, too. $\endgroup$ – Thomas Andrews Nov 12 '15 at 13:22
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Take $f(x)=x$ and $g(x)=x+\dfrac{1}{x}$

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

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