What is the limit of f(x)=4x^2−6/4x^2−1 as x approaches infinity?
A) 1
B) 3
C) 4
D) 6
E) -4
To find the limit of a function as x approaches infinity, we can examine the highest power terms in the numerator and denominator. In this case, the highest power terms are the x^2 terms.
The limit of a function as x approaches infinity is determined by the leading terms of the function. In the case of f(x) = (4x^2-6) / (4x^2 - 1), the highest power terms are both x^2.
To find the limit as x approaches infinity, we can divide all terms in the numerator and denominator by the highest power of x, which in this case is x^2.
Dividing both the numerator and denominator by x^2, we get:
f(x) = [(4x^2-6) / x^2] / [(4x^2-1) / x^2]
Simplifying further, we have:
f(x) = (4 - 6/x^2) / (4 - 1/x^2)
Now, as x approaches infinity, 1/x^2 becomes arbitrarily close to zero, and we can disregard it. Thus, the expression simplifies to:
f(x) = (4 - 0) / (4 - 0) = 4/4 = 1
Therefore, as x approaches infinity, the limit of f(x) is equal to 1.
So, the correct answer is A) 1.