Identify the indeterminate form and evaluate the limit using l'Hopital's Rule.
The limit as x approaches infinity of ln(x+1)/logbase2 of x

Question

Identify the indeterminate form and evaluate the limit using l'Hopital's Rule. 
The limit as x approaches infinity of ln(x+1)/logbase2 of x

psymon · Answer

Well, if x is approaching infinity, then straight away we have theindeterminant form infinity over infinity. Since we have the function in an undeterminant form, this satisfies the condition in which we can use l'hopitals rule. Do you know the derivatives of natural log and of logs with bases other than e?

anonymous · Answer

Yeah I know the derivative of logs

psymon · Answer

Alright, so just take the derivative of the numerator and then divide it by the derivative of the denominator and let's see what we get.

anonymous · Answer

so for the bottom the derivative would be 1 / xln2 ?

psymon · Answer

Correct.

anonymous · Answer

do i do the reciprocal of the bottom derivative in order to multiply it by the top derivative ?

psymon · Answer

Right, so the derivative for the bottom would basically flip and multiply.

anonymous · Answer

so after i get $$\lim_{x \rightarrow \infty}\frac{ x \ln 2 }{ x+1 }$$ what do I do ?

psymon · Answer

Well, we still have an indeterminant form. This means we can just use l'hopitals again.

anonymous · Answer

I use product rule for the top right ?

psymon · Answer

No need. ln2 is just a constant. So you take the derivative of it in the same way you would like 2x.

anonymous · Answer

so the limit would just be ln2 ?

psymon · Answer

Yep, thats all it would be :3

anonymous · Answer

Thank you :)

psymon · Answer

Sure ^_^