What is the limit of sin(1/x) as x approaches 0?

1 Answer
John D.
Jun 22, 2017

The limit does not exist.

Explanation:

To understand why we can't find this limit, consider the following:

We can make a new variable #h# so that #h = 1/x#.

As #x -> 0#, #h -> oo#, since #1/0# is undefined. So, we can say that:

#lim_(x->0)sin(1/x) = lim_(h->oo)sin(h)#

As #h# gets bigger, #sin(h)# keeps fluctuating between #-1# and #1#. It never tends towards anything, or stops fluctuating at any point.

So, we can say that the limit does not exist. We can see this in the graph below, which shows #f(x) = sin(1/x)#:

graph{sin(1/x) [-2.531, 2.47, -1.22, 1.28]}

As #x# gets closer to #0#, the function fluctuates faster and faster, until at #0#, it is fluctuating "infinitely" fast, so it has no limit.

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