anonymous
  • anonymous
find limt lim x approaches infinity 1/(5x+4)
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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anonymous
  • anonymous
0
anonymous
  • anonymous
denomiator gets bigger, numerator stays one. think of \[\frac{1}{1000000}=.000001\]
anonymous
  • anonymous
bigger the denominator gets, the smaller the fraction.

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anonymous
  • anonymous
ok so anything that deals with infinity could possibly be zero
anonymous
  • anonymous
well it depends on the function, but it certainly could be 0 in the limit.
anonymous
  • anonymous
ok. i have another one simliar but mor complex
anonymous
  • anonymous
shoot
anonymous
  • anonymous
\[ \lim \rightarrow \infty (1- x-x^2)/(3x^2-4)\]
anonymous
  • 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.
anonymous
  • 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
anonymous
  • anonymous
what could be easier?
anonymous
  • 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
anonymous
  • anonymous
not a thing. lets do a simple example: \[lim_{x->\infty}\frac{x^3+9x}{2x^2+x^2}\]
anonymous
  • anonymous
ok
anonymous
  • anonymous
and lets let x = 100 which is not even that big
anonymous
  • anonymous
ok cool
anonymous
  • anonymous
the numerator is \[100^3+9\times 100=1000900\]
anonymous
  • anonymous
the denominator is 2010000
anonymous
  • anonymous
the ratio is \[\frac{1000900}{2010000}\]
anonymous
  • anonymous
right and the denomator would be 2010000....
anonymous
  • anonymous
which is certainly not exactly \[\frac{1}{2}\]
anonymous
  • anonymous
ok..
anonymous
  • 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
anonymous
  • anonymous
there are way down there in the decimal places.
anonymous
  • anonymous
i need a lot of practice on this function
anonymous
  • 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
anonymous
  • anonymous
very very close to 1/2
anonymous
  • 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
anonymous
  • anonymous
i see becuase you can basically pick any number in the infinity column and it would still be the coeffiencet
anonymous
  • anonymous
now if the degree of the denominator is bigger than the numerator, then the limit is 0
anonymous
  • anonymous
think of \[\frac{10^3}{10^5}\]
anonymous
  • anonymous
numerator has degree 3, denom has degree 5 and this number is small. it is .01
anonymous
  • anonymous
it would be zero because the 10^5 and large
anonymous
  • anonymous
last possibility is that the degree of the numerator is higher. in this case the limit is infinty
anonymous
  • anonymous
yes you are right.
anonymous
  • anonymous
now think of degree of the numerator bigger. for example \[\frac{10^5}{10^2}=1000\]
anonymous
  • 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)
anonymous
  • 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
anonymous
  • anonymous
that is it!
anonymous
  • anonymous
ok no i see thanks again!!
anonymous
  • anonymous
welcome!

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