solve 3n^2-5n+4/4n^2+7n-1 as n approaches infinity
If your question mean:
lim (3n²-5n+4) / (4n²+7n-1)
n→∞
Then:
Divide numerator and denominator by n²
lim n→∞ (3 n²/n²- 5n/n²+4/n²)/(4n²/n²+7n/n²-1/n²)=
lim n→∞ (3-5/n+4/n²) / (4+7/n-1/n²) =
(Now Quotient Rule)
lim (3-5/n+4/n²)
n→∞
______________
lim (4+7/n-1/n²)
n→∞
When n→∞ then:
5 / n →0
4 / n² →0
7 / n →0
and
1 / n² →0
So:
lim (3-5/n+4/n²)
n→∞
______________ =
lim (4+7/n-1/n²)
n→∞
3 + 0 + 0
__________=
4 + 0 + 0
3 / 4
lim n→∞ (3n²-5n+4) / (4n²+7n-1) = 3 / 4
To solve the expression (3n^2 - 5n + 4) / (4n^2 + 7n - 1) as n approaches infinity, we need to analyze the highest power of n in both the numerator and denominator.
Looking at the numerator, the highest power of n is n^2. Similarly, in the denominator, the highest power of n is also n^2.
Now, we can simplify the expression by dividing each term by the highest power of n. This process is known as dividing every term by n^2.
(3n^2 - 5n + 4) / (4n^2 + 7n - 1)
= (3 - 5/n + 4/n^2) / (4 + 7/n - 1/n^2)
As n approaches infinity, the terms involving 1/n and 1/n^2 become negligible compared to the other terms.
Therefore, we can disregard these terms and simplify the expression:
(3 - 5/n + 4/n^2) / (4 + 7/n - 1/n^2)
≈ (3 - 0 + 0) / (4 + 0 - 0)
= 3 / 4
So, as n approaches infinity, the value of the expression (3n^2 - 5n + 4) / (4n^2 + 7n - 1) tends towards 3/4.
To solve the expression as n approaches infinity, we need to determine the behavior of the terms with the highest powers of n in the numerator and denominator.
In the given expression:
Numerator: 3n^2 - 5n + 4
Denominator: 4n^2 + 7n - 1
As n approaches infinity, we can ignore the lower power terms and focus on the terms with the highest powers of n.
In the numerator, the highest power term is 3n^2, and in the denominator, the highest power term is 4n^2.
Now, we divide both the numerator and denominator by the highest power of n, which is n^2, to get:
Numerator: 3 - (5/n) + (4/n^2)
Denominator: 4 + (7/n) - (1/n^2)
As n approaches infinity, the terms (5/n) and (7/n) go to zero because any number divided by infinity is essentially zero. The term (4/n^2) and (-1/n^2) also approach zero as n approaches infinity.
Hence, the expression simplifies to:
Numerator: 3
Denominator: 4
Therefore, as n approaches infinity, the expression converges to 3/4.