Problem 1 - The Calculus with Analytic Geometry

Question

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ f(x)=\frac{4}{x-5} $$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 5 \); Horizontal asymptote at \( y = 0 \).

Step 1: Identify the Vertical Asymptote

Vertical asymptotes occur where the function is undefined. For the function \( f(x) = \frac{4}{x-5} \), set the denominator equal to zero and solve for \( x \). \( x - 5 = 0 \) which gives \( x = 5 \). Therefore, there is a vertical asymptote at \( x = 5 \).

Step 2: Identify the Horizontal Asymptote

Horizontal asymptotes are found by evaluating the limits of \( f(x) \) as \( x \) approaches infinity. Since the degree of the numerator (0) is less than the degree of the denominator (1), \( \lim_{{x \to \infty}} f(x) = 0 \) and \( \lim_{{x \to -\infty}} f(x) = 0 \). Hence, the horizontal asymptote is \( y = 0 \).

Step 3: Sketch the Graph

To sketch the graph, plot the vertical asymptote at \( x = 5 \) and the horizontal asymptote at \( y = 0 \). Since \( f(x) = \frac{4}{x-5} \), as \( x \) approaches 5 from the left, \( f(x) \) approaches \( -\infty \), and as \( x \) approaches 5 from the right, \( f(x) \) approaches \( +\infty \). As \( x \) moves towards \( \pm \infty \), \( f(x) \) approaches 0.

Key Concepts

Vertical Asymptote

Understanding vertical asymptotes is key when analyzing rational functions. Vertical asymptotes occur where the function becomes undefined, i.e., where the denominator equals zero. For the function \( f(x) = \frac{4}{x-5} \), we find the vertical asymptote by setting the denominator to zero and solving for \( x \):
\( x - 5 = 0 \), which gives us \( x = 5 \).
This indicates a vertical asymptote at \( x = 5 \).

Vertical asymptotes are essentially lines that the graph will approach but never cross.
The behavior of the function near these asymptotes can show rapid changes to positive or negative infinity.

Horizontal Asymptote

Horizontal asymptotes tell us how the function behaves as \( x \) approaches infinity or negative infinity. For the function \( f(x) = \frac{4}{x-5} \), the degree of the numerator (0) is less than the degree of the denominator (1).
This means that as \( x \) approaches infinity or negative infinity, \( f(x) \) approaches 0.

Hence, the horizontal asymptote is \( y = 0 \).
Horizontal asymptotes give insights into the end behavior of a function.

For our function, at both extremes of the \( x \text{-axis} \), the graph levels off towards the \( y \text{-axis} \) at 0.

Graph Sketching

Sketching the graph of a rational function involves understanding where the function has vertical and horizontal asymptotes.
For \( f(x) = \frac{4}{x-5} \):

• Draw the vertical asymptote at \( x = 5 \).
• Draw the horizontal asymptote at \( y = 0 \).

The graph will approach but never cross these lines.
As \( x \text{ approaches} \ 5 \text{ from the left} \), \( f(x) \text{ approaches} \ -\text{∞} \).
As \( x \text{ approaches} \ 5 \text{ from the right} \), \( f(x) \text{ approaches} \ +\text{∞} \).
As \( x \text{ approaches} \ \text{±∞} \), \( f(x) \text{ approaches} \ 0 \).

This behavior creates a hyperbolic curve opening towards the vertical asymptote.
Always pay attention to the critical points and the behavior near asymptotes to accurately sketch the graph.