Series Expansion & l'Hopital's rule

[stuart4512]

I'm preparing for my exam and have stumbled across this question. I understand how to execute the l'hopital's rule part of this but I just can't get there. I have no idea how to approach this in order to get a suitable series expansion to the 4th degree of x.

View attachment 3664

Thank you

 

[topsquark]

I'm preparing for my exam and have stumbled across this question. I understand how to execute the l'hopital's rule part of this but I just can't get there. I have no idea how to approach this in order to get a suitable series expansion to the 4th degree of x.

http://mathhelpboards.com/attachments/calculus-10/3664-series-expansion-amp-lhopitals-rule-2012b3-png

Thank you

For i) you just state the definition of coth(x).

For ii) you need to find the Maclaurin series for e^{x} and e^{-x}.
then plug those expansions into the coth(x) formula.

-Dan