Problem with Limits using L'Hospital's Rule

[CharlesL]

Homework Statement

Determine the limit of

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

Differentiate

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

lim $$\psi$$$$\rightarrow$$1 ln y = 2/($$\psi$$³+3$$\psi$$²+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