L'Hospital Rule: Solving with (a) or (b)?

[jack1234]

[SOLVED] About l'hopital rule

I have two solutions for a question about limit
http://tinyurl.com/2pknkb

May I know is (a) correct, or (b)?
What is the reason for the other to be wrong?

Note:
(a) just directly applies l'hopital rule to numerator
(b) is to reduce the numerator using logarithm formula before using l'hopital rule

 

[D H]

You didn't calculate the derivative correctly for (b). You need to apply the chain rule. If you had computed it correctly you would have arrived at the same answer as for (a).

\frac d{dx}(\ln(x+1)-\ln(2)) = \frac d {dx} \ln\left(\frac{x+1} 2\right) = \frac 2 {x+1} \; \frac d {dx}\frac {x+1} 2 = \frac 2 {x+1} \, \frac 1 2 = \frac 1 {x+1}

 

[jack1234]

I see, thank you very much!