Complete the following steps for the given functions. 


a. Find the slant asymptote of $f$.


$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

Verified step by step guidance

1. Identify that the function \( f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{x^2 + 1} \) is a rational function where the degree of the numerator (3) is one more than the degree of the denominator (2). This indicates the presence of a slant (oblique) asymptote.
2. To find the slant asymptote, perform polynomial long division of the numerator \( 4x^3 + 4x^2 + 7x + 4 \) by the denominator \( x^2 + 1 \).
3. Divide the leading term of the numerator \( 4x^3 \) by the leading term of the denominator \( x^2 \) to get the first term of the quotient, which is \( 4x \).
4. Multiply the entire divisor \( x^2 + 1 \) by \( 4x \) and subtract the result from the original numerator to find the new polynomial.
5. Repeat the division process with the new polynomial until the degree of the remainder is less than the degree of the divisor. The quotient obtained (ignoring the remainder) represents the equation of the slant asymptote.

Key Concepts

Slant Asymptote

A slant (or oblique) asymptote occurs when the degree of the numerator of a rational function is exactly one higher than the degree of the denominator. To find it, perform polynomial long division on the function. The quotient (ignoring the remainder) gives the equation of the slant asymptote, which describes the behavior of the function as x approaches infinity.

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It involves dividing the leading term of the numerator by the leading term of the denominator, multiplying the entire denominator by this result, and subtracting it from the numerator. This process is repeated until the degree of the remainder is less than that of the divisor.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit various behaviors, including vertical and horizontal asymptotes, depending on the degrees of the numerator and denominator. Understanding the properties of rational functions is crucial for analyzing their limits and asymptotic behavior, particularly as x approaches infinity or specific values.