Question

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.

lim (tan(7x))^x
x~>0

Answer 1

You know that lim sinu/u = 1
so, lim tanu/u = 1 (because lim cosu = 1)

So, lim tan(7x)/x
= lim (tan(7x)*7)/(7x)
= 7 lim tanu/u where u7x
= 7

or, using L'Hospital,

lim tan(7x)/x
= lim (7 sec^2(7x))/1
= 7