Calculating the of a is an essential skill in calculus. An oblique asymptote, also known as a asymptote, is the line that a function approaches as x tends to positive or negative infinity. Unlike horizontal or vertical , oblique asymptotes are represented by linear functions rather than constants. In this article, we will explore the steps to the oblique asymptote of a given function.

To calculate the oblique asymptote, we need to perform polynomial long division on the function. This process involves dividing the numerator polynomial by the denominator polynomial and obtaining a quotient and remainder. The quotient represents the oblique asymptote, while the remainder accounts for any differences between the function and the asymptote as x approaches infinity.

Let’s consider an example to illustrate the process. Suppose we have the rational function f(x) = (3x^2 + 5x + 2) / (x – 2). Our goal is to find the oblique asymptote.

Step 1: Perform polynomial long division. Divide the numerator (3x^2 + 5x + 2) by the denominator (x – 2). The result would be:

3x + 11 (Quotient)
___________________________
x – 2 | 3x^2 + 5x + 2

– (3x^2 – 6x)
___________
11x + 2
– (11x – 22)
___________
24

Step 2: Once you have the quotient (3x + 11) and the remainder (24), the oblique asymptote can be written as y = 3x + 11.

Hence, in our example, the oblique asymptote of the function f(x) = (3x^2 + 5x + 2) / (x – 2) is y = 3x + 11.

It’s important to note that not all rational functions have an oblique asymptote. For a rational function to have an oblique asymptote, the degree of the numerator polynomial must be exactly one greater than the degree of the denominator polynomial. If the degree of the numerator polynomial is less than the degree of the denominator polynomial, then the function will have a horizontal asymptote. If the degree of the numerator polynomial is greater than the degree of the denominator polynomial, then the function will have neither a horizontal nor oblique asymptote.

Furthermore, if the quotient obtained from polynomial long division is a constant, this means the function has a horizontal asymptote instead of an oblique asymptote. In such cases, the constant value of the quotient represents the horizontal asymptote.

To summarize, calculating the oblique asymptote involves polynomial long division to obtain the quotient and remainder. The quotient represents the oblique asymptote, which is a linear function, while the remainder accounts for any differences between the function and the asymptote as x approaches infinity. Understanding how to determine whether a function has an oblique asymptote or not is crucial in analyzing and graphing rational functions, as it provides valuable insights into their behavior.

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