limit of x as it approaches infinity (x^2+sinx)/(x^2+1)?
HELP please provide a clear and detailed explanation
- YirmiyahuLv 77 years agoFavorite Answer
Look at the numerator. It fluctuates between x^2 - 1 and x^2 + 1 because sinx oscillates between -1 and +1.
Observe that the limit of (x^2 + 1) / (x^2 + 1) = 1 for all values of x.
Now look at (x^2 - 1) / (x^2 + 1) = (x^2 + 1 - 2) / (x^2 + 1) because x^2 - 1 = x^2 + 1 - 2.
(x^2 + 1 - 2) / (x^2 + 1) = (x^2 + 1) / (x^2 + 1) - 2/(x^2 + 1) = 1 - (2/(x^2 + 1))
As x approaches infinity, 1 - (2/(x^2 + 1) = 1 - 0 = 1.
So, we now have:
f(x) = (x^2 - 1) / (x^2 + 1)
g(x) = (x^2 + 1) / (x^2 + 1)
h(x) = (x^2 + sin x) / (x^2 + 1)
Observe that f(x) < h(x) < g(x)
Observe that lim f(x) = 1 = lim g(x) as x approaches infinity.
Therefore, the lim h(x) = 1 as x approaches infinity by the Squeeze Theorem
- Anonymous7 years ago
- 7 years ago
Lim x->~ (x^2+sinx)/(x^2+1) (devide all by x^2)
Lim x->~ 1+(sinx/x^2)/(1+1/x^2) = 1+0/1+0 = 1
remeber if sinx/x^2 will become 1/x
as sina/a = a/a = 1
so sinx/x^2 = x/x * 1/x =1/x which value of 1/x is 0
- 5 years ago
Apply L Hospital rule and get limit x tends to infinity (2x + cos x)/2x. Split the limit to get limit x tends to infinity 1 which is 1, plus limit x tends to infinity cos x/x which is equal to product of 0 and a finite number between -1 and 1, that is 0. Hence answer is 1
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- zymowskiLv 44 years ago
To sum all of it up basically because of fact ? is to the means of 0,?^0, we can not say that the lim as x has a tendency to ? is a million for x^(a million/x) because of fact we don't understand that's extra effective, the 0 or the ?. hence we ought to remedy it utilising L'Hospitols (L'H) rule. lim x^(a million/x) = ?^0 x->? enable y= lim x^(a million/x) x->? lny = lim ln(x^(a million/x) x->? utilising ln regulations we pull out the a million/x lny = lim (ln(x))/x = ?/? x->? Now we are able to L'H L'H lny = lim (a million/x)/a million = 0 x->? lny = 0 e^lny = y = e^0 = a million hence we are able to now genuine say that the lim x^(a million/x) = a million x->?