Evaluate the Limit

Question

Evaluate the limit below, given that

f(t) = (( 2^t+4^t)/5 )^1/t for t does not equal 0

lim t approaching positive infinity f(t) =

Answer 1

lim lnf(t) as t→∞ = lim t→∞ [ln(2^t + 4^t) - ln5}t = lim t→∞ (2^tln2 + 4^tln4)/(2^t + 4^t) = lim t →∞ (ln2 + 2^tln4)/(1 + 2^t) = lim t→∞ (ln2/2^t + ln4)/(1/2^t + 1) = (0 + ln4)/(0 + 1) = ln4;

lim f(t) t→∞ = e^ln4 = 4