The problem visible in the first image asks to analyze the rational function:

$f(x) = \frac{2x^2 + 7x + 3}{x^2 - x - 12}$

And in the second image, it asks whether the function $f(x)$ has any vertical asymptotes, and to justify the answer using limits.

Step 1: Factoring the Denominator

The vertical asymptotes occur where the denominator of the function is equal to zero, as long as these points do not cancel out with factors in the numerator.

We start by factoring the denominator:

$x^2 - x - 12 = (x - 4)(x + 3)$

Step 2: Identifying Vertical Asymptotes

Vertical asymptotes occur when the denominator equals zero and the numerator is non-zero at those points. The denominator is zero at:

$x - 4 = 0 \quad \text{or} \quad x + 3 = 0$

This gives us two potential asymptotes at $x = 4$ and $x = -3$.

Step 3: Verifying the Asymptotes Using Limits

To confirm that these are indeed vertical asymptotes, we evaluate the limits of $f(x)$ as $x$ approaches 4 and -3 from both directions.

1. As $x \to 4^+$ and $x \to 4^-$:

   $\lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} \frac{2x^2 + 7x + 3}{(x - 4)(x + 3)} = \infty \quad \text{or} \quad -\infty$

   Similarly for $x \to 4^-$.

2. As $x \to -3^+$ and $x \to -3^-$:

   $\lim_{x \to -3^+} f(x) = \lim_{x \to -3^+} \frac{2x^2 + 7x + 3}{(x - 4)(x + 3)} = \infty \quad \text{or} \quad -\infty$

   Similarly for $x \to -3^-$.

Conclusion

There are vertical asymptotes at $x = 4$ and $x = -3$.

Would you like me to expand further on limits or asymptotes, or explore other properties of this rational function? Here are five related questions to consider:

1. What happens to the function $f(x)$ at infinity (horizontal asymptote analysis)?
2. How do you find the holes in the graph of a rational function, if they exist?
3. Can we factor the numerator $2x^2 + 7x + 3$, and does it cancel out with any factors in the denominator?
4. What is the behavior of the function around its vertical asymptotes?
5. How does one use limits to find horizontal asymptotes in a rational function?

Tip: Always simplify the rational function by factoring and canceling common terms before analyzing asymptotes and limits.