Distinct answers for two equivalent expressions?

[musicgold]

Hi,
This is not exactly a homework problem. This is something I wonder all the time.

1. Homework Statement
We all know that the following expressions give different answers, even though the expressions are equivalent. My question is why does that happen. Why two expressions that are the same have different results.

Homework Equations

$\frac {lim}{x \rightarrow \infty} \ \frac {x+1}{x} =$ Indeterminant

$\frac {lim}{x \rightarrow \infty} \ 1 + \frac {1}{x} =$ 1

The Attempt at a Solution

Is there a problem with the basic mathematical axioms that is creating this situation? Why should it matter how I write the expression?

 

[Doc Al]

Homework Equations

$\frac {lim}{x \rightarrow \infty} \ \frac {x+1}{x} =$ Indeterminant

While the fraction may be an indeterminate form, the limit is not indeterminant. (See L'Hospital's Rule.)

 

[fresh_42]

$\dfrac{x+1}{x}=1+\dfrac{1}{x} \neq x+\dfrac{1}{x}$

 

[Doc Al]

$\frac {lim}{x \rightarrow \infty} \ x + \frac {1}{x} =$ 1

I just noticed what you wrote here. Note that this is not equivalent at all to the other fraction. (@fresh_42 beat me to it.)

I assume this was just a typo!

 

[musicgold]

$\dfrac{x+1}{x}=1+\dfrac{1}{x} \neq x+\dfrac{1}{x}$

Thanks. I fixed it.

 

[musicgold]

While the fraction may be an indeterminate form, the limit is not indeterminate. (See L'Hospital's Rule.)

The L Hospital's rule address some other issue - how to take the limit when we have an indeterminate form.

It doesn't address the basic question - Why two equivalent equations result in different results.

 

[fresh_42]

The L Hospital's rule address some other issue - how to take the limit when we have an indeterminate form.

It doesn't address the basic question - Why two equivalent equations result in different results.

$\lim_{x \to \infty} \dfrac{x+1}{x} = \lim_{x\to \infty}\left( 1+\dfrac{1}{x} \right) = 1$
There is nothing "indeterminant".

 

[Doc Al]

It doesn't address the basic question - Why two equivalent equations result in different results.

But they don't have different results!

 

[RPinPA]

The L Hospital's rule address some other issue - how to take the limit when we have an indeterminate form.

It doesn't address the basic question - Why two equivalent equations result in different results.

They aren't different results. They are different methods of proof of the same result.

When you say the form is "indeterminate", that's a statement that you don't yet know what the limit is. It is not a statement about the limit. The result is 1. If you evaluated $(x + 1)/x$ for an increasing sequence of $x$, you would find that for any value of $\epsilon > 0$, the value of $(x + 1)/x$ was within $\epsilon$ of 1 for all $x$ large enough. And that, by definition, means the limit is 1.

 

[haruspex]

how to take the limit when we have an indeterminate form.

The only reason I can see why you consider it indeterminate is that you are doing:
$\underset{x \rightarrow \infty}{lim} \frac {x+1}{x} =\frac {\underset{x \rightarrow \infty}{lim} x+1}{\underset{x \rightarrow \infty}{lim} x}=\frac \infty\infty$.
The trouble with that step is that it throws away crucial information, namely, that the two limit processes are to occur at the same 'rate', i.e. the values of x increase in synch. That loss of information leads to the indeterminacy.