What is the Limit of xln(x) - x as x Approaches 0?

[blessedcurse]

Homework Statement

What is the value of xln(x)-x when x=0?

Homework Equations

I'm assuming you do L'Hopital's

The Attempt at a Solution

I'm assuming you factor out the x, leaving:

x(ln(x)-1)

but that's still not in the form of $$\frac{\infty}{\infty}$$ or $$\frac{0}{0}$$

Would you do:

$$lim_{x\rightarrow\infty}$$$$\frac{(ln(x)-1)}{x^{(-1)}}$$

=$$lim_{x\rightarrow\infty}$$$$\frac{(1/x)}{1}$$

=0

??

 

[gabbagabbahey]

Homework Statement

What is the value of xln(x)-x when x=0?

Homework Equations

I'm assuming you do L'Hopital's

The Attempt at a Solution

I'm assuming you factor out the x, leaving:

x(ln(x)-1)

but that's still not in the form of $$\frac{\infty}{\infty}$$ or $$\frac{0}{0}$$

Would you do:

$$lim_{x\rightarrow\infty}$$$$\frac{(ln(x)-1)}{x^{(-1)}}$$

=$$lim_{x\rightarrow\infty}$$$$\frac{(1/x)}{1}$$

=0

??

$$\lim_{x\to 0} x\ln(x)-x=(\lim_{x\to 0} x\ln(x))-(\lim_{x\to 0} x)$$