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Let $f,g$ be two functions defined on $[a, \infty] $ for some $a$ .

Give a counterexample to the following statements:

1) If $ \lim_{x\to \infty } (f(x)-g(x)) = \infty $ , then $ \lim_{x\to \infty } (\frac{f(x)}{g(x)} )= \infty $

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

3) If $ \lim_{x\to \infty } |\frac{f(x)}{g(x)} |= \infty $, then $ \lim_{x\to \infty } |f(x)-g(x)| = \infty $

Thanks in advance

3 Answers 3

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$f(x)=1$,

$1)$ $g(x)=-x$. $2,3) \ g(x)=\frac{1}{x}.$

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Hints: For 1) you need $f$ to grow faster than $g$, but not too much faster. For 2) you can let $g$ get small. For 3, if your answer to 2 had both $f$ and $g$ positive, the same one will work.

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  1. $f(x)=x^2+x+1; \;\; g(x)=x^2.$
  2. $f(x)=\dfrac{1}{x};\;\; g(x)=\dfrac{1}{x^2}.$
  3. $f(x)=\dfrac{1}{x};\;\; g(x)=\dfrac{1}{x^2}.$