I need help on calc 2

Question

find limits

 

 lim
x→0   x^rootx

Answer 1

lim_(x→0)(x^√x) has the indeterminate form 0⁰

 

Let y = x^√x

 

Then lny = √xlnx = lnx/(1/√x)

 

We will first find the limit of lny as x approaches 0 and then use that limit to find the given limit.

 

lim_x→0(lny) = lim_(x→0)[lnx/(1/√x)] ← has the form -∞/∞

 

                = lim_(x→0)[(1/x)/(-½x^-3/2)] (By L'Hopital's Rule)

 

                = -2lim_(x→0)(√x) = 0

 

So, lim_(x→0)(x^√x) = lim_(x→0)y

 

                      = lim_(x→0)e^lny = e⁰ = 1