# lHop2 - l'Hospital's Rule A Classic Example 1 Let's compute...

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l'H os pital's Rule: A Classic Example ( 29 1 Let's compute the limit: lim 1 Notice, if I plug in the " ", we, formally, get: 1 1 lim 1 1 1 0 1 But recognize, the base isn't quite 1. x x x x x x → ∞ → ∞ + + = + = + = And the exponent, of course isn't even a real number, so what's going on? Well, any number slightly larger than 1, to a large power, would expand to . Conversely, any number slightly smaller than 1 wou ld shrink away to zero. And of course, an value, taken to any power, even a very large one, is still 1. So as it stands, it appears we have an . Now, is there any way to change th exact indeterminate form is indeterminate form into one we know how to deal with? Well, since the problem is with the exponent, let's bring it down with a ln-operation. 1 If we let lim 1 then ln( ) ln lim x x x y x y → ∞ = + = 1 1 1 1 limln 1 lim ln 1 Again, plugging in our " ", then

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lHop2 - l'Hospital's Rule A Classic Example 1 Let's compute...

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