Identify all horizontal and vertical asymptotes of the graph of the function. f(x)=x^2/x^2-4
I know the vertical asymptotes are x=2 and x=-2, but how do I find the horizontal asymptotes?
Just consider what happens when x gets large. At x = 10^100, say, all but the highest powers of x get swamped, so f(x) just looks like
x^2/x^2 = 1
So, y=1 is the horizontal asymptote
Or, you can do a polynomial division, to see that
x^2/(x^2-4) = 1 + 4/(x^2-4)
as x gets huge, 4/(x^2-4) goes to zero, leaving f(x) = 1.
To find the horizontal asymptotes of a function, you need to analyze the behavior of the function as x approaches positive or negative infinity.
For the given function f(x) = x^2 / (x^2 - 4), we can determine the horizontal asymptotes by looking at the degrees of the numerator and denominator polynomials.
First, let's consider what happens as x approaches infinity (positive values). As x becomes very large, the higher degree terms dominate the function. Both the numerator (x^2) and the denominator (x^2 - 4) have the same highest degree, which is 2.
When the degrees of the numerator and denominator are the same, the ratio of their coefficients gives the horizontal asymptote. In this case, the coefficient of x^2 in the numerator is 1, and the coefficient of x^2 in the denominator is also 1.
Therefore, as x approaches infinity, the function approaches a horizontal asymptote of y = 1.
Next, let's consider what happens as x approaches negative infinity. Again, the ratio of the coefficients of the highest power terms will give the horizontal asymptote.
In the given function, both the numerator (x^2) and the denominator (x^2 - 4) approach the same value as x approaches negative infinity. Therefore, the horizontal asymptote is also y = 1.
To summarize, the graph of the function f(x) = x^2 / (x^2 - 4) has two horizontal asymptotes: y = 1 as x approaches both positive and negative infinity.