Question

need help with limit problem

Answer 1

okay so my question is why is it dividing by n?

Answer 2

i thought you divided by the value with the largest power

Answer 3

guys pls help me out in my question

Answer 4

@amistre64

Answer 5

@IrishBoy123

Answer 6

Next you should be asking why is it dividing by the term with largest power

Answer 7

i dont understand

Answer 8

Notice that "n" is indeed the largest power in the denominator, but clearly you're missing the key point, dividing by "n" is not really necessary here

Answer 9

Look at the expression
\[\dfrac{n^3}{n+8}\]

what happens to this term as "n" becomes large ?
which one grows faster, numerator or denominator ?

Answer 10

the denominator

Answer 11

are you saying "n+8" grows faster than "n^3" ?

Answer 12

plugin n=10, 100 etc and see which one is growing faster

Answer 13

no i meant the numerator

Answer 14

Okay what about the value of expression as "n" gets large ?

Answer 15

it goes to infinity right

Answer 16

As you can see the function f(x) = x^3/(x+8) is increasing without any bound as x increases, so the corresponding sequence diverges

Answer 17

They are dividing top and bottom by "n" so that it becomes easy for you to see the same

Answer 18

\[a_n = \dfrac{n^3}{n+8} = \dfrac{n^2}{1+8/n}\]

"plugin" \(n = \infty\), the expression becomes \[ \dfrac{\infty^2}{1+8/\infty} = \dfrac{\infty^2}{1+0} = \infty\]

Answer 19

i think i understand better now

Answer 20

In general :

For any rational function, \(f(x) = \dfrac{P(x)}{Q(x)}\) ,
as \(x\to\infty\), we have \(f(x)\to\pm\infty\) if the degree of \(P(x)\) is greater than the degree of \(Q(x)\)