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

Luke Alvarez · Accepted Answer

I'll try to give an example below.

Example 1: #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. Second example: #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. To obtain the outcome, we can apply the squeeze theorems on the left and right. The value lim inf = lim sup indicates that the limit is also that value.

Scarlett Allison · Answer

undefined The limit of a function represents the value that the function approaches as the input approaches a certain value. The limsup (limit superior) and liminf (limit inferior) are used to describe the behavior of a function when it oscillates or fluctuates near a certain point.

The limsup of a function is the largest limit point that the function can reach as the input approaches a certain value. It is the supremum (or maximum) of all the limit points.

The liminf of a function is the smallest limit point that the function can reach as the input approaches a certain value. It is the infimum (or minimum) of all the limit points.

To illustrate with an example, let's consider the function f(x) = sin(1/x). As x approaches 0, this function oscillates between -1 and 1 infinitely many times. The limit of f(x) as x approaches 0 does not exist because the function does not approach a single value.

However, we can find the limsup and liminf of f(x) as x approaches 0. The limsup is 1, as the function reaches its maximum value of 1 infinitely many times. The liminf is -1, as the function reaches its minimum value of -1 infinitely many times.

In summary, the limit of a function represents the value it approaches, while the limsup and liminf describe the largest and smallest limit points respectively, when a function oscillates or fluctuates near a certain point.