Question

How do I calculate this limit?

lim of [(3x-1)^20 * (2x+3)^10] / [(2x-1)^30]
as x->infinity

(without L'Hospital! thanks.)

Answer 1

Hmm this probably isn't rigorous enough, but here's how I approach it.\[\Large\lim_{x\to\infty}\frac{(3x-1)^{20}(2x-1)^{10}}{(2x-1)^{30}}\quad\text{~}\quad\lim_{x\to\infty}\frac{(3x)^{20}(2x)^{10}}{(2x)^{30}}\]Since it's an infinite limit, the only thing that will really affect it is the leading terms.

Answer 2

And then beyond that you can pull out the constants to further see what's going on.\[\Large ~\quad \frac{3^{20}2^{10}}{2^{30}}\lim_{x\to\infty}\frac{x^{20}x^{10}}{x^{30}}\]

Answer 3

Thanks a lot! :D

Answer 4

c: