00:01
Okay, so we want to find the limit of this function here.
00:04
And when you put, when x approaches a from the left, you get, basically you get natural log of one on the bottom, which is zero.
00:12
On the other fraction, you get one over zero.
00:14
Basically, you get like infinity minus infinity, which is an indeterminate form, but we can't do anything with that.
00:22
So it's not an indeterminate form where we can use l 'hospital's rule.
00:26
So what i'm gonna do is i'm gonna get a common denominator.
00:29
I'm gonna write this as x minus a minus the natural log of x over a all over x minus a times the natural log of x over a.
00:47
And it's the limit.
00:48
I just made it one fraction as we approach, x approaches a from the left.
00:58
Can't put, you know, and again, you can't put a in there.
01:01
You get zero on top, zero on the bottom, but now this comes out to be zero over zero, which is a indeterminate form that we can now do something with.
01:14
So now i'm gonna use l 'hospital's rule.
01:16
I'm gonna take the derivative of the numerator.
01:18
A, i'm assuming is just a constant.
01:20
So it's gonna be a derivative of x is just one.
01:24
The derivative of the natural log of x over a is gonna be one over x over a times one a, times, i don't wanna say it.
01:39
Okay, so it'll be minus.
01:42
So the derivative of the natural log of x over a.
01:50
I'm gonna do one thing first that i think is gonna make it.
01:55
I'm gonna use the rule of logs to make this a little bit easier.
01:58
I'm gonna write that natural log of x over a as a natural log of x minus the natural log of a.
02:04
It's gonna make things a lot easier.
02:09
It still comes out to be, so if i do the limit as x approaches a from the left of x minus a minus the natural log of x minus the natural log of a, right? rules of logs.
02:27
And then on the bottom, i have x minus a times natural log of x minus the natural log of a.
02:37
Still equals zero over zero.
02:40
But now when we distribute this minus sign in, it's gonna be a plus natural log of a.
02:48
Not that that matters.
02:50
Because when i take the derivative of the top, i get one, the derivative of a is, the constant is just zero, minus one over x plus the derivative of the natural log of a is zero because it's constant.
03:08
So plus zero.
03:09
So this is what's on top...