Question

How do I find the limit or show whether this does not converge

Answer 1

\[\frac{ 3 \times 10^{n+1} -5^{n+2} }{ 2^{n+1} -4 \times 10^{n} }\]

Answer 2

@Loser66
my first thought was to divide top and bottom by a^n
but they all have different powers

any suggestions?

Answer 3

I think it is divergent because 5/2 > 1

Answer 4

\[\dfrac{30 *10 ^n -25*5^n}{2*2^n-4*10^n}\]

Answer 5

But \(10^n = 2^n*5^n\), so, we can factor to get \(\left(\dfrac{5}{2}\right)^{n+1}* something\)

Answer 6

Notice that \(10^n - 5^n\) approaches \(10^n\) for large values of \(n\).

Answer 7

As \(n\rightarrow \infty\), this first element goes to infinity also.
that is what I think :)

Answer 8

Basically you can ignore all the lower bases in top and bottom :

\[\lim\limits_{n\to\infty}\frac{ 3 \times 10^{n+1} -5^{n+2} }{ 2^{n+1} -4 \times 10^{n} }=\lim\limits_{n\to\infty}\dfrac{3\times 10^{n+1}}{-4\times 10^n}\]

Answer 9

:)

Answer 10

Alternatively, you may divide top and bottom by 5^n to reach the same thing..

Answer 11

thank you guys :) I am just not used to so many different powers

Answer 12

Btw it converges to -15/2

Answer 13

thank you!