Anyone can explain to me what's the difference between "limit", "limsup" and "liminf" of a function? It would be helpful to explain with concrete example.

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Answer 1 · 2017-02-15T15:15:18

I'll try to give an example below.

Example1 :

$f(x) = sin(1/x)$ as $xrarr0$

Every deleted $epsilon$ ball around $0$ has supremum $1$ , so

$lim_(xrarr0) "sup" f(x) = 1$

Every deleted $epsilon$ ball around $0$ has infimum $-1$ , so

$lim_(xrarr0) "inf" f(x) = -1$

As we know $lim_(xrarr0) sin(1/x)$ does not exist.

Example 2:

$g(x) = xsin(1/x)$ as $xrarr0$

Every deleted $epsilon$ ball around $0$ has supremum $epsilon$ , so

$lim_(xrarr0) "sup" f(x) = lim_(epsilonrarr0) epsilon = 0$

Every deleted $epsilon$ ball around $0$ has infimum $-epsilon$ , so

$lim_(xrarr0) "inf" f(x) = lim_(epsilonrarr0) - epsilon = 0$

We know that $lim_(xrarr0) xsin(1/x)= 0$ , for two reasons.

We can use the squeeze theorem on the left and right to get the result.

If we have lim sup = lim inf, then that value is also the limit.