To find the limit of $g(t) = \frac{t + 4}{|t + 4|}$ as $t$ approaches $-4$, let's analyze the behavior of the function as $t$ gets close to $-4$ from both the left and the right.

1. Consider the limit as $t$ approaches $-4$ from the right ($t \to -4^+$):

• When $t > -4$, the expression inside the absolute value $|t + 4|$ is positive. Therefore, $|t + 4| = t + 4$.
• The function becomes: $g(t) = \frac{t + 4}{t + 4} = 1$
• So, as $t$ approaches $-4$ from the right: $\lim_{{t \to -4^+}} g(t) = 1$

2. Consider the limit as $t$ approaches $-4$ from the left ($t \to -4^-$):

• When $t < -4$, the expression inside the absolute value $|t + 4|$ is negative. Therefore, $|t + 4| = -(t + 4)$.
• The function becomes: $g(t) = \frac{t + 4}{-(t + 4)} = -1$
• So, as $t$ approaches $-4$ from the left: $\lim_{{t \to -4^-}} g(t) = -1$

3. Evaluate the limit:

The left-hand limit and the right-hand limit are not equal:

• $\lim_{{t \to -4^+}} g(t) = 1$
• $\lim_{{t \to -4^-}} g(t) = -1$

Since the left-hand limit and right-hand limit are different, the limit does not exist as $t$ approaches $-4$.

So, the correct conclusion is:

$\lim_{{t \to -4}} g(t) \text{ does not exist.}$

Would you like to explore any specific part of this explanation in more detail, or do you have another question?

Here are some related questions:

1. How does the limit change if $t$ approaches a different value where the expression is non-zero?
2. Can you explain why limits might not exist at points where functions have discontinuities?
3. What are the conditions under which absolute value functions create discontinuities?
4. How would you graph $g(t) = \frac{t + 4}{|t + 4|}$ and interpret the results?
5. What is the difference between left-hand and right-hand limits, and how do they determine if a limit exists?

Tip: When dealing with limits involving absolute values, always check the behavior of the function from both the left and the right side of the point in question, especially if the expression inside the absolute value can change sign.