what is the oblique asymptote for the rational function: x^2+13x+42/x+7

To find the oblique asymptote of a rational function, you divide the numerator by the denominator using either polynomial division or synthetic division.

Given the function:

\[
f(x) = \frac{x^2 + 13x + 42}{x + 7}
\]

Performing polynomial division:

1. Divide \(x^2\) by \(x\), which gives \(x\).
2. Multiply \(x\) by \(x + 7\), which gives \(x^2 + 7x\).
3. Subtract \(x^2 + 7x\) from \(x^2 + 13x + 42\), leaving \(6x + 42\).
4. Divide \(6x\) by \(x\), which gives \(+6\).
5. Multiply \(6\) by \(x + 7\), which gives \(6x + 42\).
6. Subtract \(6x + 42\) from \(6x + 42\), leaving a remainder of 0.

Thus, the quotient is:

\[
f(x) = x + 6
\]

Therefore, the oblique asymptote is the line:

\[
y = x + 6
\]

Would you like more details on the division process or any related questions?

Here are 5 related questions for deeper understanding:
1. What are the conditions for a rational function to have an oblique asymptote?
2. How is an oblique asymptote different from a horizontal asymptote?
3. What happens if the degree of the numerator is less than the degree of the denominator?
4. Can a rational function have both a horizontal and an oblique asymptote?
5. How does polynomial division help in finding the behavior of rational functions?

**Tip**: Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator.

What is the oblique asymptote for the rational function: x^2 + 13x + 42 / x + 7?

Learn how to find the oblique asymptote of the rational function x^2 + 13x + 42 / x + 7 using polynomial division. This step-by-step guide covers key concepts in rational functions and asymptotic behavior, suitable for students in Grades 10-12.

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