To find the vertical and horizontal asymptotes of the rational function $f(x) = \frac{5}{x^2}$, let's break it down step by step:

1. Vertical Asymptotes:

Vertical asymptotes occur when the denominator of the function is equal to zero because the function tends to infinity near these points.

For $f(x) = \frac{5}{x^2}$, the denominator is $x^2$. Setting the denominator equal to zero: $x^2 = 0$ Solving for $x$: $x = 0$ Therefore, there is a vertical asymptote at $x = 0$.

2. Horizontal Asymptotes:

Horizontal asymptotes describe the behavior of the function as $x$ approaches infinity ($x \to \infty$) or negative infinity ($x \to -\infty$).

• As $x \to \infty$, $f(x) = \frac{5}{x^2} \to 0$ because the denominator becomes very large.
• Similarly, as $x \to -\infty$, $f(x) = \frac{5}{x^2} \to 0$ as well.

Thus, the horizontal asymptote is: $y = 0$

Final Answer:

• Vertical asymptote: $x = 0$
• Horizontal asymptote: $y = 0$

Would you like any further clarification or details on these concepts?

5 Relative Questions:

1. What happens to the function near the vertical asymptote $x = 0$?
2. Can a function have more than one vertical asymptote?
3. How does the degree of the denominator affect vertical asymptotes?
4. What is the behavior of rational functions as $x \to \infty$?
5. How do horizontal asymptotes differ from oblique asymptotes?

Tip:

Always check both $x \to \infty$ and $x \to -\infty$ when finding horizontal asymptotes, as they describe the end behavior of the function.