Indeterminate Forms

[confused_07]

Can someone explain this to me? I understand the form "0/0" and "inf/inf" as x-> a, but I do not understand the "0/inf" or "inf/0" forms. Example:

lim x-> inf x ln [ (x-1)/(x+1)]

The text book says this is a "0/inf" form. How? What am I not seeing?

 

[confused_07]

Ok.... when you apply that to the equation, it makes it into a "0/0" form. Got it.

What's the differnce between a "0 * inf" form and a "inf * 0" form?

The reason I ask is that the next section talks about the "0^0", "inf^0", and "1^inf" forms, so I want to understand the first ones before I even try to attempt to learn these.

Oh, yeh, thanks for the help....

 

[galactus]

If you write it as \(\displaystyle \L\\\frac{ln(\frac{x-1}{x+1})}{\frac{1}{x}}\)

L'Hopital, taking derivative of top and bottom gives:

\(\displaystyle \L\\\frac{-2x^{2}}{(x-1)(x+1)}\)

\(\displaystyle \L\\-2\lim_{x\to\infty}\frac{x^{2}}{(x-1)(x+1)}\)

Rewrite:

\(\displaystyle \L\\-2\lim_{x\to\infty}\frac{x^{2}}{x^{2}-1}\)

Divide top and bottom by x^2:

\(\displaystyle \L\\-2\lim_{x\to\infty}\frac{1}{1-\frac{1}{x^{2}}}\)

Now, as x appraoches infinity, the limit approaches -2

 

[confused_07]

Got it... thanks!!!