To find vertical asymptotes of a rational function, follow these steps:

Definition:

A vertical asymptote occurs when the function approaches infinity or negative infinity as the input value $x$ approaches a specific value. This typically happens at the zeros of the denominator (where the denominator equals zero), provided these points do not also make the numerator zero (resulting in a hole instead of an asymptote).

Steps to Find Vertical Asymptotes:

1. Write the Function: Represent the rational function in the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ is the numerator and $Q(x)$ is the denominator.

2. Set the Denominator Equal to Zero: Solve $Q(x) = 0$. The solutions to this equation are the potential vertical asymptotes.

3. Check for Common Factors: If the numerator $P(x)$ and denominator $Q(x)$ have common factors, cancel them out. The corresponding zeros of these common factors indicate holes, not vertical asymptotes.

4. Identify the Vertical Asymptotes: Any remaining solutions to $Q(x) = 0$ after canceling common factors are the locations of the vertical asymptotes.

Example:

Consider the function $R(x) = \frac{x^2 - 1}{x^2 - 4}$.

1. Write the function: $R(x) = \frac{(x - 1)(x + 1)}{(x - 2)(x + 2)}$.

2. Set denominator to zero: $x^2 - 4 = 0$, so $x = \pm 2$.

3. Check for common factors: No common factors exist between numerator and denominator.

4. Vertical asymptotes: The vertical asymptotes are at $x = 2$ and $x = -2$.

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Do you want further details or examples? Here are five related questions you might explore:

1. What happens if both the numerator and denominator have the same factor at a given $x$?
2. How do vertical asymptotes differ from horizontal asymptotes?
3. How can you graph a rational function to visualize vertical asymptotes?
4. How do removable discontinuities affect the presence of vertical asymptotes?
5. How do you find vertical asymptotes for functions with trigonometric terms in the denominator?

Tip: Always simplify the rational function first to distinguish between holes and vertical asymptotes!