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So I'm having a bit of difficulty with two of these L'Hospital's Rule problems...

The first:

[tex]\mathop {\lim }\limits_{x \to \infty} (\sqrt{x^2 + x} - x)[/tex]

So when you have an indefinite form [tex]\infty - \infty[/tex], you've got to turn it into a product indefinite form, usually something like [tex]\infty * 0[/tex]

So I take that,

[tex]\mathop {\lim }\limits_{x \to \infty} (\sqrt{x^2 + x} - x) = \mathop {\lim }\limits_{x \to \infty} x(\frac{\sqrt{x^2 +x}}{x} - 1)[/tex]

From there, I try to take the limit of the fraction in the parenthesis, but end up going in circles with it. After I differentiate it the first time and get another indefinite form, I differentiate again and from there, it loops. I keep getting the reciprocal of what I started with. I know that the answer is supposed to come out to be 1/2, but I have no idea how to get there. Any ideas?

The second one I'm having difficulty with is,

[tex]\mathop {\lim }\limits_{x \to 1+} lnx tan(\pi x/2)[/tex]

I'm getting a similar problem with this one. No matter which way I go when I try to turn this one into a fraction, I either get the wrong answer (-1), or it just gets continually more complex. ...The answer here is supposed to be [tex]-2/\pi[/tex]

Any pointers?

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# Homework Help: L'Hopital's Rule, 2 Confusing ones

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