The function((x^2 + 7x + 14)^(1/2)) - x
has one horizontal asymptote at y=?
To determine the horizontal asymptote of the function f(x) = ((x^2 + 7x + 14)^(1/2)) - x, we need to examine the behavior of the function as x approaches positive and negative infinity.
To find the horizontal asymptote, we need to calculate the limit of the function as x approaches infinity and negative infinity:
1. Let's start by finding the limit as x approaches infinity. We substitute infinity into the function:
lim(x->∞) ((x^2 + 7x + 14)^(1/2)) - x
To evaluate this limit, we need to consider the highest power of x in the function, which is x in this case. Now, we need to determine the highest power's coefficient — in this case, it is 1.
2. Divide every term in the equation by x, keeping the terms in the same order:
lim(x->∞) ([(x^2)/x + (7x)/x + (14)/x]^(1/2)) - [x/x]
Simplifying this expression:
lim(x->∞) ([x + (7/x) + (14/x)]^(1/2)) - 1
Since 7/x and 14/x tend to 0 as x approaches infinity, our expression simplifies further:
lim(x->∞) ([(x + 0 + 0)]^(1/2)) - 1
lim(x->∞) (x^(1/2)) - 1
lim(x->∞) √x - 1
As x approaches infinity, √x also approaches infinity, so the limit becomes:
lim(x->∞) √x - 1 = ∞ - 1 = ∞
This means that as x approaches infinity, the value of the function also approaches positive infinity.
3. Now, let's find the limit as x approaches negative infinity. We substitute negative infinity into the function:
lim(x->-∞) ((x^2 + 7x + 14)^(1/2)) - x
Following the same steps as before, we simplify the expression:
lim(x->-∞) ([(x^2)/x + (7x)/x + (14)/x]^(1/2)) - [x/x]
lim(x->-∞) ([x + (7/x) + (14/x)]^(1/2)) - 1
Since 7/x and 14/x tend to 0 as x approaches negative infinity, our expression simplifies further:
lim(x->-∞) ([(x + 0 + 0)]^(1/2)) - 1
lim(x->-∞) (x^(1/2)) - 1
lim(x->-∞) √x - 1
As x approaches negative infinity, √x becomes imaginary. The square root of a negative number is not defined in the real number system. Therefore, the limit is undefined.
In summary, the function ((x^2 + 7x + 14)^(1/2)) - x has no horizontal asymptote, as the limit is ∞ as x approaches positive infinity and the limit is undefined as x approaches negative infinity.