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How to find the vertical asymptote and horizontal asymptote of this kind of rational function

To find the vertical and horizontal asymptotes of the rational function given:

\[
f(x) = \frac{2}{2x + 5}
\]

### **Vertical Asymptote:**

A vertical asymptote occurs where the denominator of the function is equal to zero (since division by zero is undefined). Set the denominator equal to zero and solve for \(x\):

\[
2x + 5 = 0
\]

Solving for \(x\):

\[
2x = -5 \implies x = -\frac{5}{2}
\]

So, the vertical asymptote is at:

\[
x = -\frac{5}{2}
\]

### **Horizontal Asymptote:**

To determine the horizontal asymptote, we compare the degrees of the numerator and the denominator:

- The degree of the numerator (which is a constant \(2\)) is 0.
- The degree of the denominator (which is \(2x + 5\)) is 1.

Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at:

\[
y = 0
\]

### **Summary:**

- **Vertical Asymptote:** \(x = -\frac{5}{2}\)
- **Horizontal Asymptote:** \(y = 0\)

Would you like more details or have any questions?

#### Related Questions:
1. What happens to the function as \(x\) approaches the vertical asymptote?
2. How do we find the slant (oblique) asymptote if the degrees are the same or the numerator's degree is higher?
3. What is the domain of this function?
4. How do we determine if there are any holes in the graph?
5. How does the function behave as \(x\) approaches infinity?

**Tip:** When analyzing rational functions, always start by checking for vertical asymptotes and horizontal/slant asymptotes to understand the general shape of the graph.

Learn how to find the vertical and horizontal asymptotes of the rational function f(x) = 2 / (2x + 5) using algebraic techniques. Understand the concepts of vertical and horizontal asymptotes, the method of solving for them, and their significance in graphing rational functions.

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