Find the Asymptotes f(x)=(2x-5)/(x-6)

Solution:

Step 1:

Determine the value of $x$ that causes $\frac{2x-5}{x-6}$ to be undefined, which is $x=6$.

Step 2:

Examine the general form of a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

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

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

• When $n>m$, there is no horizontal asymptote, but there may be an oblique asymptote.

Step 3:

Identify the values of $n$ and $m$ for the given function.

• $n=1$
• $m=1$

Step 4:

Since $n$ is equal to $m$, calculate the horizontal asymptote using the formula $y=\frac{a}{b}$, where $a=2$ and $b=1$.

• $y=2$

Step 5:

Conclude that there is no oblique asymptote since the degree of the numerator is not greater than the degree of the denominator.

Step 6:

Summarize the asymptotes of the function.

• Vertical Asymptotes: $x=6$
• Horizontal Asymptotes: $y=2$
• No Oblique Asymptotes

Step 7:

Knowledge Notes:

To find the asymptotes of a rational function, one must first understand the different types of asymptotes:

1. Vertical Asymptotes: These occur at values of $x$ where the function is undefined, typically where the denominator is zero. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.

2. Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and the denominator ($m$) of the rational function:

   • If $n<m$, 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.

3. Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. To find an oblique asymptote, divide the numerator by the denominator using polynomial long division or synthetic division.

In the given problem, the function $f(x)=\frac{2x-5}{x-6}$ has a vertical asymptote at $x=6$ because the function is undefined when $x=6$. The degrees of the numerator and denominator are both 1 ($n=m=1$), so the horizontal asymptote is found using the leading coefficients of the numerator and denominator, resulting in $y=\frac{2}{1}=2$. Since the degree of the numerator is not greater than the degree of the denominator, there are no oblique asymptotes.