The tutor works a limit using the log trick and l’Hôpital’s rule.

Example: Evaluate lim_n→∞n^(1/n)

Solution:

First, imagine

y=lim_n→∞n^(1/n)

Next, we take the logarithm of both sides:

lny=lnlim_n→∞n^(1/n)

Now, because both lny and n^(1/n) are continuous n→∞, we can change their order on the right side:

lny=lim_n→∞lnn^(1/n)

Using the log rule about exponent-to-multiple gives

lny=lim_n→∞(1/n)lnn

or

lny=lim_n→∞lnn/n

This limit has the ∞/∞ form, which means, by l’Hôpital’s Rule, we can take the derivative of the numerator and denominator separately, then take the limit of that result:

lny=lim_n→∞(1/n)/1=lim_n→∞(1/n)

So,

lny = 0

We take the exponential of both sides:

e^(lny) = e^0

to arrive at

y=1

Recalling that y=lim_n→∞(1/n), we realize that

lim_n→∞n^(1/n)=1

HTH:)

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus. Toronto:
   D.C. Heath and Company, Ltd., 1986.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.