Find the limit. Use I'Hospital's Rule where appropriate.

Question

lim x → 0 (x4^x) / (4^x − 1)

Answer 1

The given limit has the indeterminate form 0/0. So, L'Hopital's Rule is applicable.

Applying the rule, the given limit is equivalent to:

lim_x→0 [ (4^x + xln4(4^x)) / (ln4(4^x)) ] = lim_x→0 [(1 + xln4) / (ln4) ] = 1 / ln4

Answer 2

lim x → 0 (x4^x) / (4^x − 1) = lim x → 0 (x4^x)' / (4^x − 1)' = 'lim x → 0 (4^x + x4^xln4) / (4^xln4) = 4⁰/(4⁰ln4) = 1/ln4.

We applyed here L'Holpital'rule because we have indenterminate form (0/0)