Determine the horizontal aymptotes of the function f(x)=3x^2/x^2+1
To determine the horizontal asymptotes of the given function f(x) = 3x^2 / (x^2 + 1), we need to examine the behavior of the function as x approaches positive and negative infinity.
As x approaches positive or negative infinity, the term with the highest degree in the denominator becomes dominant. In this case, the term is x^2.
Since both the numerator and denominator have the same highest degree term (x^2), we can compare their coefficients to determine the horizontal asymptotes.
The coefficient of x^2 in the numerator is 3, while the coefficient of x^2 in the denominator is 1.
Therefore, the horizontal asymptote of the function is y = 3/1, which simplifies to y = 3.
So, the horizontal asymptote of the function f(x) = 3x^2 / (x^2 + 1) is y = 3.
To determine the horizontal asymptotes of the function f(x) = 3x^2 / (x^2 + 1), we can examine the end behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, both the numerator and denominator of the fraction approach infinity. Therefore, we can divide both the numerator and denominator by the highest power of x, which in this case is x^2.
Dividing the numerator 3x^2 by x^2, we get:
(3x^2 / x^2) = 3
Dividing the denominator x^2 + 1 by x^2, we get:
(x^2 + 1 / x^2) = (1 + 1 / x^2)
As x approaches infinity, (1 / x^2) approaches zero. Therefore, we have:
(1 + 1 / x^2) approaches 1 + 0 = 1
Thus, as x approaches positive infinity, the function approaches the value 3 / 1 = 3.
Similarly, as x approaches negative infinity, both the numerator and denominator of the fraction approach infinity. By dividing them both by x^2, we find:
(3x^2 / x^2) = 3
Therefore, as x approaches negative infinity, the function also approaches the value 3.
Hence, the horizontal asymptote of the function f(x) = 3x^2 / (x^2 + 1) is y = 3.