Do I even need to use L'Hopital's Rule for this

tjohn101
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Homework Statement


Evaluate the limit analytically if necessary using L’Hopital’s rule:
lim x->0 (1+x)1/x


Homework Equations





The Attempt at a Solution


Well, I can get the thing equal to 1 if I just plug in zero, so do I need to use L'hopital's? This whole thing is very confusing...
 
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How can you 'plug in 0'?? 1/0 is undefined. Besides the limit isn't 1. I suggest you use l'Hopital.
 
To be honest, it's late and I've been doing this stuff for 9 hours straight. It didn't even cross my mind that 1/0 would be undefined.
 
Gotcha. So just take the log and use l'Hopital on the result.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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