Question

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

Answer 1

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.

Final Answer