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.

Question

4886 views around the world

You can reuse this answer
Creative Commons License

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.