Thread: Indeterminate Forms and L'Hospital's Rule

1. Indeterminate Forms and L'Hospital's Rule

1. lim x->1 (x/x-1 - 1/ln x)
This is what I got so far...
=> lim x->1 (x-1)/(x-1 ln x)
=> lim x->1 d/dx (x-1)/d/dx (x-1 ln x)
=> lim x->1 1/x ln x .....I got stuck here
I check the book and the answer is 1/2 but how?

2. lim x->infinity ((squareroot x^2+x)-x)
This is what I got so far...
=> lim x->infinity d/dx (x^2+x^1/2 -x)
=> lim x->infinity d/dx (-x)/d/dx (x^2+x^-1/2) ...I got stuck here
The answer from the book is 1/2 but how?

2. Originally Posted by jsu03
1. lim x->1 (x/x-1 - 1/ln x)
This is what I got so far...
=> lim x->1 (x-1)/(x-1 ln x)
I'm assuming the top line is $\frac{x}{x-1}-\frac{1}{\ln(x)}$

Then you made a mistake combining these two fractions. I don't think you need to combine these fractions though, since the limit at x=1 yields $\infty - \infty$, which is an acceptable indeterminate form.