## anonymous 5 years ago find limt lim x approaches infinity 1/(5x+4)

1. anonymous

0

2. anonymous

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

3. anonymous

bigger the denominator gets, the smaller the fraction.

4. anonymous

ok so anything that deals with infinity could possibly be zero

5. anonymous

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

6. anonymous

ok. i have another one simliar but mor complex

7. anonymous

shoot

8. anonymous

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

9. anonymous

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.

10. anonymous

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

11. anonymous

what could be easier?

12. anonymous

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

13. anonymous

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

14. anonymous

ok

15. anonymous

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

16. anonymous

ok cool

17. anonymous

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

18. anonymous

the denominator is 2010000

19. anonymous

the ratio is $\frac{1000900}{2010000}$

20. anonymous

right and the denomator would be 2010000....

21. anonymous

which is certainly not exactly $\frac{1}{2}$

22. anonymous

ok..

23. anonymous

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

24. anonymous

there are way down there in the decimal places.

25. anonymous

i need a lot of practice on this function

26. anonymous

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

27. anonymous

very very close to 1/2

28. anonymous

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

29. anonymous

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

30. anonymous

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

31. anonymous

think of $\frac{10^3}{10^5}$

32. anonymous

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

33. anonymous

it would be zero because the 10^5 and large

34. anonymous

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

35. anonymous

yes you are right.

36. anonymous

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

37. anonymous

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)

38. anonymous

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

39. anonymous

that is it!

40. anonymous

ok no i see thanks again!!

41. anonymous

welcome!