A limit using l'hopital's rule

I am wassim from Montreal ( student) and I have this problem in an econometrics course

I need to know how to solve the:
limits of (x ( to the power lamda) -1 )/LAMDA when lamda tends to zero ( the answer is that the functional form is ln x) and I still don't know how using hopital rule leads to this answer.

Hi Wassim,

As lamda tends to zero, x^lamda tends to 1 and hence the numerator and denominator both tend to zero. Thus this problem is a candidate for l'hopital's rule.

My assumption is that your difficulty is differentiating x^lamda with respect to lamda. If you don't recall how to differentiate this form ( a constant raised to a variable power) you can proceed as follows.

Let y = x^lamda and take the natural log of both sides to get ln(y) = ln(x^lamda) = lamda ln(x). Now differentiate both sides with respect to lamda.
y'/y = ln(x) and hence y' = y ln(x) = x^lamda ln(x).

Now apply l'hopital's rule to your limit and consider the new limit

Thus

Cheers,
Harley Go to Math Central