Find the Asymptotes 7-8/x

Solution:

Step 1:

Identify the values for which $7 - \frac{8}{x}$ does not exist. This occurs when $x = 0$.

Step 2:

Examine the general form of a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$, where $n$ represents the degree of the polynomial in the numerator and $m$ represents the degree of the polynomial in the denominator. The following rules apply to determine horizontal asymptotes:

1. If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.

2. If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.

3. If $n > m$, there is no horizontal asymptote; instead, there may be an oblique asymptote.

Step 3:

Determine the values of $n$ and $m$. For the given function, $n = 0$ and $m = 1$.

Step 4:

Since $n = m$, apply the rule to find the horizontal asymptote, which is $y = \frac{a}{b}$. Here, $a = 7$ and $b = 1$, so the horizontal asymptote is $y = 7$.

Step 5:

An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.

Step 6:

Compile the list of all asymptotes for the function:

• Vertical Asymptotes: $x = 0$
• Horizontal Asymptotes: $y = 7$
• No Oblique Asymptotes

Step 7:

The process is complete.

Knowledge Notes:

To find the asymptotes of a function, we need to understand the behavior of the function as it approaches certain critical points or infinity. Asymptotes are lines that the graph of a function approaches but never touches.

• Vertical Asymptotes occur at values of $x$ that make the function undefined, typically where the denominator of a rational function is zero.

• Horizontal Asymptotes are found by comparing the degrees of the numerator and denominator of a rational function. If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y = 0$. If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If $n > m$, there is no horizontal asymptote.

• Oblique Asymptotes may occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, we perform long division to find the equation of the oblique asymptote.

In the given problem, the function $7 - \frac{8}{x}$ is a rational function with a constant numerator and a linear denominator. The function is undefined at $x = 0$, which is where the vertical asymptote lies. Since the degrees of the numerator and denominator are equal ($n = m = 1$), we find the horizontal asymptote by taking the ratio of the leading coefficients, which in this case is $y = 7$. There is no oblique asymptote because the degree of the numerator is not greater than the degree of the denominator.