Limit Problem

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December 30th 2012, 04:46 AM
marijakopljar

Limit Problem

Please,
if we have:
lim(x to 1) ((x/(x-1)-1/lnx)
then we get:

lim(x to 1) ((xlnx-x+1)/(xlnx-lnx))
then we take x=1:

lim(x to1)((ln(1)+1-1) / (ln(1)-ln(1))
we get 0/0
and can apply LH:

but NOW:
How do we get NOW lim(x to1)((xlnx)/(xlnx+x-1)???
What is the inversion doing here?
Many thanks!
December 30th 2012, 06:34 AM
skeeter

Re: Limit Problem

$\lim_{x \to 1} \frac{x}{x-1} - \frac{1}{\ln{x}}$

$\lim_{x \to 1} \frac{x\ln{x}}{(x-1)\ln{x}} - \frac{x-1}{(x-1)\ln{x}}$

$\lim_{x \to 1} \frac{x\ln{x} - (x-1)}{(x-1)\ln{x}}$

$\lim_{x \to 1} \frac{x\ln{x} - x + 1}{(x-1)\ln{x}}$

L'Hopital ...

$\lim_{x \to 1} \frac{x \cdot \frac{1}{x} + \ln{x} - 1}{(x-1)\cdot \frac{1}{x} + \ln{x}}$

$\lim_{x \to 1} \frac{\ln{x}}{1 - \frac{1}{x} + \ln{x}}$

multiply numerator and denominator by $x$ ...

$\lim_{x \to 1} \frac{x\ln{x}}{x - 1 + x\ln{x}}$

L'Hopital again ...

$\lim_{x \to 1} \frac{1+ \ln{x}}{2+\ln{x}} = \frac{1}{2}$
December 30th 2012, 11:40 AM
marijakopljar

Re: Limit Problem

Please...ther start all the problems...

in the fifth line how did u get x*(1/x)+lnx-1?

it should be d/dx xlnx-x+1, shouldn't it?

p.s. re U using latex?
December 30th 2012, 11:45 AM
zachd77

Re: Limit Problem

In line 5, that's the same thing as the derivative of x*lnx-x+1. If you look at the definition of L'Hopital's Rule, it implies differentiation of the numerator and the denominator. Therefore, line 5 shows the derivative of the numerator and the denominator from line 4. Line 5 simply results from using L'Hopital's Rule.
December 30th 2012, 11:47 AM
skeeter

Re: Limit Problem

Quote:

Originally Posted by marijakopljar

Please...ther start all the problems...

in the fifth line how did u get x*(1/x)+lnx-1?

it should be d/dx xlnx-x+1, shouldn't it?

p.s. re U using latex?

have you not learned the product rule for derivatives ?

$\frac{d}{dx}[{\color{red}x \ln{x}} - x + 1] = {\color{red}x \cdot \frac{1}{x} + \ln{x} \cdot 1} - 1 = 1 + \ln{x} - 1 = \ln{x}$

... and yes, the script is Latex.
December 30th 2012, 02:20 PM
marijakopljar

Re: Limit Problem

Thank You, that is exactly what I missed!!!
December 31st 2012, 01:57 AM
marijakopljar

Re: Limit Problem

Dear All!
Please, could anyone help me with this:
Find number a is element of R so that the function f(x)=a; x<=0
( x^1/2-1)/(x^1/3-1); x>0

is continuous in R?

R: a=3/2

p.s. how to get latex here? I tried to write by tags, but it just showed the source, what I wrote, and not the nice latex text!
December 31st 2012, 05:42 AM
skeeter

Re: Limit Problem

Quote:

Originally Posted by marijakopljar

Dear All!
Please, could anyone help me with this:
Find number a is element of R so that the function f(x)=a; x<=0
( x^1/2-1)/(x^1/3-1); x>0

is continuous in R?

R: a=3/2

p.s. how to get latex here? I tried to write by tags, but it just showed the source, what I wrote, and not the nice latex text!

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