Find the horizontal asymptote of the graph of the function. (If an answer does not exist, enter DNE.)
f(x) = x^3 − 3x^2 + 5x + 1/
x^2 − 5x + 3
since the numerator is of higher degree than the denominator, there is no horizontal asymptote.
As x gets huge, the lower powers become insignificant, and the fraction approaches
x^3/x^2 = x
So, there is a slant asymptote, but ho horizontal asymptote. The graph is at
http://www.wolframalpha.com/input/?i=%28x^3+%E2%88%92+3x^2+%2B+5x+%2B+1%29%2F%28+x^2+%E2%88%92+5x+%2B+3%29
To find the horizontal asymptote of a function, we need to examine the behavior of the function as x approaches positive and negative infinity.
First, let's check the degrees of the numerator and denominator. The degree of the numerator is 3, and the degree of the denominator is 2.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0 (the x-axis).
If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at y = a/b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
If the degree of the numerator is greater than the degree of the denominator, then the function does not have a horizontal asymptote.
In this case, the degree of the numerator (3) is greater than the degree of the denominator (2). Therefore, the function does not have a horizontal asymptote.
Hence, the answer is DNE (Does Not Exist).