00:01
So we're given a function, f of x, which is a piecewise function.
00:04
And the piecewise function just tells us that, well, if our input is zero, then we output zero.
00:10
And if our input is non -zero, then we use the component part, x squared times sine of 1 over x as what we're going to be plugging that non -zero x into.
00:20
And the question we're being asked is whether or not the graph of this piecewise function has a tangent at the origin.
00:28
And another way, i think maybe a better way of thinking about this, is thinking about whether or not f of x has a derivative at the point x not equals zero.
00:42
So let's think about this, x not equals zero.
00:46
Now, my claim is that there does exist a tangent at the origin if there's a derivative existing at x not equals zero.
00:53
And the definition to check whether or not the derivative exists at x not equals zero is this definition where x not is just the point of interest that we're looking at.
01:14
So f of x not plus h minus f of x not over h.
01:17
This is just the standard definition for a derivative.
01:21
And if this comes out to be some sort of finite number, then we'll say that the limit exists and if the limit exists, then f prime of 0 will be whatever that result is, and that will mean that a tangent does exist at the origin...