The ideas and solutions are as follows. Compare yourself!

Thinking: Obviously, with the important limit lim (x→∞) [1+(1/x)] (x) = e, we can solve it by fitting.

Original limit = lim (x→∞) [1+(3a)/(x-a)] (x/3)

= lim(x→∞)[ 1+(3a)/(x-a)]^[(x-a)/(3a)*(3a)/(x-a)*(x/3)]

= lim(x→∞){ 1+(3a)/(x-a)]^[(x-a)/(3a)} ^[(3a)/(x-a)*(x/3)]

=e^{lim(x→∞)[(3a)/(x-a)*(x/3)]}

= e (a) = 8 = east (ln8)

∴a = ln8

Therefore:

a=3ln2