Question

How do you do this question with or without a calculator?

Answer 1

Hello from MrBillDoesMath!

Answer:

1

Discussion:

Problem: Evaluate ln(x) / (x-1) as x approaches 1.

As ln(1) = 0 and (1-1) = 0 this ratio is an 0/0 indeterminate form. Let's use L'Hopitals theorem:

....as x approaches 1,

ln(x) / ( x- 1) = derivative of (ln (x)) / derivative of (x-1) =

1/x / 1 = 1/x

As x approaches 1, 1/x approaches 1/1 or 1

Regards,

MrB

Answer 2

Explanation:

The idea is not usual, but once you have seen the answer, I think you will grasp the principle so well that you will understand the concept much better when you hit the idea in higher mathematics which you will. Take a much easier example of the same thing.

What is 0/0? What does it mean.

Suppose it does equal something. What is that something? Write it like this.

• a * 0 = 0 What is the value of a?
• Could it be 5? Sure, why not. 5*0 is 0.
• Could it be 18? Sure, why not. 18*0 = 0
• Could it be pi? And again why not. Pi * 0 = 0

Having established that it can be anything at all, we turn to your problem.

ln(1) = 0 Try that on your calculator.

Now as x approaches 1, x - 1 becomes 0.

So what you have is

`(lim)/(x \to 1)\text{x - 1}=0`

Now you get 0/0 which has no definite answer by the remarks made above.