To find the limit of the expression as $x \to -7^-$, we need to evaluate the behavior of the function

$\frac{12}{x + 7}$

as $x$ approaches $-7$ from the left (indicated by the $-$ sign).

Step-by-step Solution:

1. Expression Setup: $f(x) = \frac{12}{x + 7}$

2. Approaching $x = -7$ from the Left:

   As $x$ approaches $-7$ from the left ($x \to -7^-$), the denominator $x + 7$ approaches $0$. Specifically, since $x$ is slightly less than $-7$, $x + 7$ will be a small negative number.

3. Behavior of the Function:

   The expression $\frac{12}{x + 7}$ will become very large in magnitude but negative, as dividing by a small negative number results in a large negative value.

4. Conclusion:

   As $x \to -7^-$, $\frac{12}{x + 7} \to -\infty$.

Limit Result:

$\lim_{x \to -7^-} \frac{12}{x + 7} = -\infty$

The limit of $\frac{12}{x + 7}$ as $x$ approaches $-7$ from the left is $-\infty$.

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Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. What happens to $\lim_{x \to -7^+} \frac{12}{x + 7}$?
2. How would the limit change if the numerator were negative?
3. How do you determine the limits of rational functions at vertical asymptotes?
4. What is the difference between one-sided limits and two-sided limits?
5. Can you explain what happens to $\frac{12}{x+7}$ when $x$ approaches a number other than $-7$?

Tip: Always consider the direction from which $x$ is approaching when calculating one-sided limits.