Question

find limt
lim x approaches infinity 1/(5x+4)

Answer 1

0

Answer 2

denomiator gets bigger, numerator stays one. think of
\[\frac{1}{1000000}=.000001\]

Answer 3

bigger the denominator gets, the smaller the fraction.

Answer 4

ok so anything that deals with infinity could possibly be zero

Answer 5

well it depends on the function, but it certainly could be 0 in the limit.

Answer 6

ok. i have another one simliar but mor complex

Answer 7

shoot

Answer 8

\[ \lim \rightarrow \infty (1- x-x^2)/(3x^2-4)\]

Answer 9

ok in this case you have a polynomial of degree 2 in the numerator, and a polynomial of degree 2 in the denominator. when the degrees are the same, as in this case, just take the ratio of the leading coefficients.

Answer 10

leading coefficient of the numerator is -1, and the leading coefficient of the denominator is 3 so the limit as x goes to infinity is -1/3

Answer 11

what could be easier?

Answer 12

ok..so basically you look into the coefficent of both numerator and denominator and computer it to become a -1/3...so the -4 in the denomintor and the numerator dont neccessary needs anything

Answer 13

not a thing. lets do a simple example:
\[lim_{x->\infty}\frac{x^3+9x}{2x^2+x^2}\]

Answer 14

ok

Answer 15

and lets let x = 100 which is not even that big

Answer 16

ok cool

Answer 17

the numerator is
\[100^3+9\times 100=1000900\]

Answer 18

the denominator is 2010000

Answer 19

the ratio is
\[\frac{1000900}{2010000}\]

Answer 20

right and the denomator would be 2010000....

Answer 21

which is certainly not exactly \[\frac{1}{2}\]

Answer 22

ok..

Answer 23

but you can see that it is close. you can also see that the 9x term and the x^2 term meant nothing as far as the magnitude of the numerator and denominator

Answer 24

there are way down there in the decimal places.

Answer 25

i need a lot of practice on this function

Answer 26

and that was just for 100. we are taking the limit as x -> infinity. imagine what it would look like if x was 10000000000

Answer 27

very very close to 1/2

Answer 28

this is not how it is explained in your book. i am just pointing out that the numbers show you that you will get closer and closer to 1/2

Answer 29

i see becuase you can basically pick any number in the infinity column and it would still be the coeffiencet

Answer 30

now if the degree of the denominator is bigger than the numerator, then the limit is 0

Answer 31

think of
\[\frac{10^3}{10^5}\]

Answer 32

numerator has degree 3, denom has degree 5 and this number is small. it is .01

Answer 33

it would be zero because the 10^5 and large

Answer 34

last possibility is that the degree of the numerator is higher. in this case the limit is infinty

Answer 35

yes you are right.

Answer 36

now think of degree of the numerator bigger. for example
\[\frac{10^5}{10^2}=1000\]

Answer 37

and that is just for 10! as x -> infinity this will get huge. so the rules are as follows (and very simple)
lim x-> infity p(x)/q(x)

Answer 38

if deg p > deg q, limit is infinity
if deg p < deg q limit is 0
if deg p = deg q ratio of leading coefficients

Answer 39

that is it!

Answer 40

ok no i see thanks again!!

Answer 41

welcome!