Help me solve question on limit?

Question

Please show me the step to solve this question:
http://i6.photobucket.com/albums/y248/zfc/lim.jpg

Answer 1

We want lim(n->inf)((1+2 *5^n)/(4+3* 5^n)). If we divide the numerator and the denominator by 5^n, we get

lim(n->inf) (5^(-n) + 2)/(4 * 5^(-n) + 3).

5^(-n) --> 0 as n -> oo. Therefore, as n --> oo, the numerator goes to 2 and the denominator goes to 3. It follows our limit is 2/3.

Answer 2

write 5^n as x.
if 5^n tends to infinity, then x also tends to infinity.

so the limit value becomes Lim(x->inf)((1+2x)/(4+3x))
now take x common in both the numerator and denominator:
you get (x(1/x+2))/(x(4/x+3))
the x 's get cancelled. now substitute infinity for x, you get 2/3, as anything divided by infinity returns zero.