00:01
In this question, we are asked to calculate the given limit.
00:04
First of all, note that this limit is in indeterminate form, infinity to the power of zero.
00:10
We can't calculate limits of this type directly.
00:15
And we will first rewrite 1 minus 9x to the 1 over x as, we will factor out negative 9.
00:24
Negative 9x minus 1 to the power 1 over x.
00:30
And this equals negative 1 to the 1 over x multiplied by 9x minus 1 to the 1 over x.
00:39
Now, this term here goes to 1 as x goes to infinity because 1 over x is zero and negative 1 to the zero equals to 1.
00:53
Now, let's focus on the second part.
00:57
We will calculate the limit of ln of 9x minus 1 to the 1 over x.
01:06
And the reason for factoring out the negative sign was, we cannot take limit of ln 1 minus 9x.
01:16
Because when x goes to infinity, 1 minus 9x becomes negative.
01:21
And we cannot take a ln of a negative number.
01:24
But here, 9x minus 1 is positive when x goes to infinity.
01:28
So we are fine here.
01:31
Now, by using properties of logarithms, we'll factor out 1 over x.
01:34
We'll get the limit of 1 over x times ln of 9x minus 1.
01:40
This equals limit of ln of 9x minus 1 divided by x...