The equation for the vertical asymptotes comes from setting the denominator of the rational function equal to zero because vertical asymptotes occur where the function becomes undefined.

From the given rational function:

$F(x) = \frac{(2x+1)(x-4)}{(x+1)(x-2)}$

The denominator is:

$(x+1)(x-2)$

Set the denominator equal to zero:

$(x+1) = 0 \quad \text{or} \quad (x-2) = 0$

This gives:

$x = -1 \quad \text{and} \quad x = 2$

Thus, the equations for the vertical asymptotes are:

$x = -1 \quad \text{and} \quad x = 2$

Would you like further clarification or additional details?

Expanding Understanding

1. What is the relationship between vertical asymptotes and limits?
2. How can you verify the presence of vertical asymptotes using a graph?
3. Why don’t vertical asymptotes occur at points where the numerator equals zero?
4. What is the effect of removing factors common to the numerator and denominator on asymptotes?
5. How do horizontal asymptotes differ from vertical asymptotes in behavior?

Tip: To ensure a function has a vertical asymptote, always check that the zero from the denominator is not canceled by a similar factor in the numerator.