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for two function $f(x),g(x)$. we consider the limit :

$$L=\lim_{x\to a}\frac{f(x)}{g(x)}$$

if: $\color{blue}{L=+\infty}$ then:$\color{blue}{f(x) \gg g(x)}$ at $\color{blue}{x=a}$($f\color{red}{\text{ grows faster than} }g$)

if : $\color{blue}{L=0}$ then: $\color{blue}{f(x) \ll g(x)}$ at $\color{blue}{x=a}$($g\color{red}{\text{ grows slower than} }f$)

if : $\color{blue}{0then: $\color{blue}{f(x) \approx g(x)}$ at $\color{blue}{x=a}$

if : $\color{blue}{\color{blue}{-\inftythen: $\color{blue}{f(x) \approx g(x)}$ at $\color{blue}{x=a}$

if : $\color{blue}{L=-\infty}$ then: $\color{blue}{f(x) \ll g(x)}$ at $\color{blue}{x=a}$

is it right??

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    Then, replace $g$ by $-g$ and use the above? – 2017-01-23
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    How you define $<<$ and $>>$.? – 2017-01-23
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    Also, tip: in $\LaTeX$, use $\gg$ and $\ll$ (`\gg,\ll`). – 2017-01-23
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    @MyGlasses It's standard notation for "is much smaller than" and "is much larger than". It's not something exactly defined, and as such it doesn't see much use in mathematics, but it's avidly used in physics. – 2017-01-23
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    Replace $f$ by $-f$. – 2017-01-23
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    You haven't defined what you mean by $\approx;$ it would be good if you did. – 2017-01-23
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    @Arthur In every case, aren't the corresponding quantities positive? (i.e., either by themselves, or because one looks at their magnitude)? These notations for functions that take negative values is bound to be confusing as hell, and most likely wrong half the time. – 2017-01-23

2 Answers 2

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You are right, as long as $f$ and $g$ are both positive around $a$. If you use absolute value signs in your "if" statements, then it is completely correct. As for the negative cases of $L$, it's the same thing, except the functions have opposite sign.

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    I edited . is it right?? – 2017-01-23
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If $L= \infty,$ then it's possible $f(x)\ll g(x).$ Example: $f(x)=-x^2, g(x) = -x.$ You have similar problems with some of the others.

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    What is true??? – 2017-01-23