00:01
Okay, here we're going to take the limit as x goes to zero of the cosine of x.
00:07
So here's the unit circle.
00:11
And at the angle zero, we have the point one zero.
00:14
And remember the cosine is the x value and the sign is the y value.
00:19
So the cosine of zero is one to the three over zero power.
00:24
So one to the infinity, which is an indeterminate form.
00:28
Okay, because there's an exponent that tells me to set it equal to y.
00:31
So that i can take the natural log of both sides.
00:36
So i have the natural log of y is the natural log of the limit.
00:43
And then in the next step, i'm going to change the order there.
00:50
Okay, because the natural log of the limit is the limit of the natural log.
00:55
So i have limit as x goes to zero.
00:59
Natural log of the cosine of x to the 3 over x squared power.
01:07
All right.
01:08
So now what i'm going to do is the reason.
01:10
The reason why i took the log of both sides is because it allows me to bring the exponent down to the front.
01:17
So now i have limit as x goes to 0, 3 over x squared times the natural log of the cosine of x.
01:29
Again, remember the natural log looks like this.
01:35
So when we plug in x equals 0, we have the cosine of 0, which is 1, and the natural log of 1, which is 0.
01:42
So now we have zero on the top and zero on the bottom.
01:46
So we can do lopetal's rule...