Limits dealing with indeterminate forms

[dwaonng]

Suppose you have one limit

$$lim_{x\rightarrow \ 0}(cos(x)/x) = \infty$$

and a second limit

$$lim_{x\rightarrow \ \infty}(x) = \infty$$

What is the first limit subtracted by the second? Is it simply indeterminate because its inf - inf?

One friend suggested I assume x=cos(y)/y for the second limit then change the second limit to look as follows:

$$lim_{x\rightarrow \ \infty}(x) =? lim_{y\rightarrow \ 0}(cos(y)/y)$$

Then can I say:
$$lim_{x\rightarrow \ 0}(cos(x)/x) - lim_{y\rightarrow \ 0}(cos(y)/y) =? 0$$ ?

 

[Hurkyl]

What is the first limit subtracted by the second? Is it simply indeterminate because its inf - inf?

I assume you are specifically asking for the difference in the values of the two limits? That difference is undefined. In particular, writing an expression denoting their difference is grammatically incorrect.

If you were trying to solve a problem and arrived at this expression, you (probably) have encountered an indeterminate form. If the problem is simply about this difference, then the answer is that the difference is undefined.


Just to make sure my point is clear, even this expression:

$$\left( \lim_{x \rightarrow +\infty} x \right) - \left( \lim_{x \rightarrow +\infty} x \right)$$

is undefined; the difference does not exist, and it would be incorrect to assert this difference is zero.

 

[dwaonng]

So you are saying that I can't manipulate the following:$$\left( \lim_{x \rightarrow +\infty} x \right) - \left( \lim_{x \rightarrow +\infty} x \right)$$

into say...

$$\left( \lim_{x \rightarrow +\infty} x - x \right)$$ ?

is that correct?

 

[Hurkyl]

So you are saying that I can't manipulate the following:


$$\left( \lim_{x \rightarrow +\infty} x \right) - \left( \lim_{x \rightarrow +\infty} x \right)$$

into say...

$$\left( \lim_{x \rightarrow +\infty} x - x \right)$$ ?

is that correct?

Correct.