What is the limit of the function as x approaches infinity?
(x^4 - 7x + 9) / (4 + 5x + x^3)
From what I know, the limit should be infinity since the greater exponent is in the numerator. However, I am only given the options: 0, (1/4), 1, 4, or Does not exist. Is there an error on the part of the answers given, or am i doing this wrong?
You are right.
as x gets big, the function looks like x^4/x^3 which is x which gets big. Infinity is correct.
To find the limit of a function as x approaches infinity, you need to analyze the behavior of the function for very large values of x.
In this case, let's denote the function as f(x) = (x^4 - 7x + 9) / (4 + 5x + x^3).
First, observe that the numerator has an x^4 term, and the denominator has an x^3 term. As x approaches infinity, the higher order terms dominate, so the behavior of the function is primarily determined by the x^4 term in the numerator and the x^3 term in the denominator.
Since the exponent is greater in the numerator, the function grows without bound as x approaches infinity, indicating that the limit should indeed be infinity. However, given the multiple-choice options presented, it appears there might be an error.
To clarify, let's perform the long division to obtain a more simplified expression:
x
______________
x^3 + 5x + 4
- (x^4 - 7x + 9)
x^4 + 5x^2 + 4x
_____________________
- 5x^2 + 4x + 9
- (- 5x^2 - 25x - 20)
_____________________
29x + 29
The long division yields a simplified expression of (x - 1) with a remainder of (29x + 29) / (x^3 + 5x + 4).
Now let's analyze the limit of this simplified expression as x approaches infinity.
As x approaches infinity, the (29x + 29) term will increase without bound, and the denominator (x^3 + 5x + 4) will also increase without bound.
In this case, the numerator grows linearly with x, while the denominator grows cubically with x. Therefore, the denominator dominates and the result approaches infinite values in the positive or negative direction.
Considering this analysis, if the options provided are 0, (1/4), 1, 4, or "Does not exist," none of them appear to be correct choices. The limit should approach infinity.
It is possible that there was an error either in the options presented or in the problem itself. Please double-check the problem and the answer choices.