Problem with Limits using L'Hospital's Rule

[CharlesL]

Homework Statement

Determine the limit of

lim \psi\rightarrow1 of \psi^(1/(\psi-1))

Homework Equations

The Attempt at a Solution

Let y = \psi^(1/(\psi-1))
ln y = ln \psi^(1/(\psi-1))

lim \psi\rightarrow1 ln y = lim \psi\rightarrow1 of (1/(\psi-1)) (ln \psi)

Differentiate

lim \psi\rightarrow1 ln y = -1/(\psi-1)² x (1/\psi)

lim \psi\rightarrow1 ln y = 2/(\psi³+3\psi²+3\psi+1)

ln y =1/4
y = e^1/4

Does e^1/4 = e?

 

[Hogger]

L'Hopital's rule works for f(x)/g(x) and then you get f'(x)/g'(x). Try rewriting the step before you differentiate as a fraction and not a product.

 

[CharlesL]

Thank you for your reply.

I wonder which is the correct solution

solution (a)

ln y = 1/(\psi-1) x ln \psi

ln y = ln \psi x (\psi-1)

ln y = 1/\psi

ln y = 1/1

y = e¹

or solution (b)

ln y = 1/(\psi-1) x ln \psi

ln y = ln \psi / (\psi-1)

ln y = 1/\psi

ln y = 1/1

y = e¹

 

[Hogger]

I did it the second way, assuming you just didn't feel like typing out that you were still dealing with limits

 

[CharlesL]

Thank you Hogger for your point outs. Appreciate it. Have a nice day



Charles