evaluate the limit

Question

lim (1+4x)^3/x

as x goes to 0

Answer 1

Is the expression on which you're taking the limit the (3/x) power of (1+4x)?  I imagine so, but it would have helped had you enclosed this quotient (exponent) in its own parentheses.

 

Attack this problem logarithmically.  lim (x→0)  [3ln(1+4x) ÷ x] gives 0/0 so we employ l'Hopital's rule;

 

lim (x→0) [3/(1+4x)•4 ÷ 1] =  12.  

 

Therefore the answer is e¹².