Question

For \(f(x)=\frac{2 x^{2}-5 x-4}{x-7},\) find all vertical asymptotes, horizontal asymptotes, and oblique asymptotes, if any.

Short answer

Vertical asymptote: \(x = 7\). No horizontal asymptote. Oblique asymptote: \(y = 2x + 9\).

Step-by-step solution

Identify Vertical Asymptotes

To find the vertical asymptotes of the function, set the denominator equal to zero and solve for x. The denominator is given by \[ x - 7 = 0 \] Therefore, the vertical asymptote occurs at \[ x = 7 \]

Identify Horizontal Asymptotes

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. The degree of the numerator (\(2x^2\)) is 2, and the degree of the denominator (\(x\)) is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Identify Oblique Asymptotes

For oblique asymptotes, observe that when the degree of the numerator is exactly one more than the degree of the denominator, the function has an oblique asymptote. Perform polynomial long division on \[ \frac{2x^2 - 5x - 4}{x - 7} \] First, divide \(2x^2\) by \(x\) to get \(2x\). Multiply \(2x\) by \(x - 7\) to get \(2x^2 - 14x\). Subtract to get \(9x - 4\). Then, divide \(9x\) by \(x\) to get \(9\). Multiply \(9\) by \(x - 7\) to get \(9x - 63\). Subtract to get \(59\).Thus, the quotient is \(2x + 9\).The oblique asymptote is given by \[ y = 2x + 9 \]

Key concepts

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero, as these points make the function undefined. To find vertical asymptotes, set the denominator equal to zero and solve for x. For the function given by: $$ f(x) = \frac{2x^2 - 5x - 4}{x-7} $$ the denominator is $$ x - 7 $$.
Set $$x - 7 = 0$$ and solving for x gives: $$ x = 7$$.
Therefore, the vertical asymptote for this function is at $$ x = 7 $$. This means as the x-value approaches 7 from either side, the function value will approach positive or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. To determine if a function has a horizontal asymptote, compare the degrees of the polynomial in the numerator and the denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $$ y = 0 $$.
• If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In our function: $$ f(x) = \frac{2x^2 - 5x - 4}{x-7} $$
The degree of the numerator (2) is greater than the degree of the denominator (1). Therefore, $$ f(x) $$ has no horizontal asymptote.

Oblique Asymptotes

Oblique or slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find the oblique asymptote, perform polynomial long division. Let's divide our function:
$$ \frac{2x^2 - 5x - 4}{x - 7} $$ by using polynomial long division:

• First, divide $$ 2x^2 $$ by $$ x $$ to get $$ 2x $$. Multiply $$ 2x $$ by $$ x-7 $$ to get $$ 2x^2 - 14x $$.
• Subtract to find the remainder: $$ (2x^2 - 5x - 4) - (2x^2 - 14x) = 9x - 4 $$.
• Next, divide $$ 9x $$ by $$ x $$ to get $$ 9 $$. Multiply $$ 9 $$ by $$ x - 7 $$ to get $$ 9x - 63 $$.
• Subtract again to find the remainder: $$ (9x - 4) - (9x - 63) = 59 $$.

The quotient from the division is $$ 2x + 9 $$, which indicates the oblique asymptote is: $$ y = 2x + 9 $$. Hence, as x approaches infinity or negative infinity, the function behaves similarly to the line $$ y = 2x + 9 $$.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, and it is necessary when you need to find oblique asymptotes. Here’s a simple set of steps to follow:

• Write the divisor and the dividend in standard form, ensuring that all terms are present (including those with zero coefficients).
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this new term of the quotient and subtract from the dividend to find the remainder.
• Repeat this process using the new remainder as the dividend until the degree of the remainder is less than the degree of the divisor.

Let's apply these steps:
Given the division of: $$ \frac{2x^2 - 5x - 4}{x - 7} $$

1. First term: $$ \frac{2x^2}{x} = 2x $$
2. Calculate: $$ 2x(x - 7) = 2x^2 - 14x $$
3. Subtract: $$ (2x^2 - 5x - 4) - (2x^2 - 14x) = 9x - 4 $$
4. Next term: $$ \frac{9x}{x} = 9 $$
5. Calculate: $$ 9(x - 7) = 9x - 63 $$
6. Subtract: $$ (9x - 4) - (9x - 63) = 59 $$

The quotient gives us $$ 2x + 9 $$, which is used to identify the oblique asymptote. This method not only helps in understanding the behavior of the function at large values of x but also provides a clear division process for complex polynomials.