Let's go through the problems step by step.

Problem 1:

You are asked to compute the following limit: $\lim_{x \to 4} \frac{x + 3}{x - 4}$

Solution:

First, substitute $x = 4$ directly into the function: $\frac{4 + 3}{4 - 4} = \frac{7}{0}$

Since the denominator approaches 0 while the numerator approaches 7, the limit tends towards infinity or negative infinity depending on the direction of the approach (from the left or the right).

Thus: $\lim_{x \to 4} \frac{x + 3}{x - 4} = \infty \text{ or } -\infty$ This needs to be broken down further in the next steps.

Problem 2a:

You are asked to find: $\lim_{x \to 4^+} \frac{x + 3}{x - 4}$

Solution:

As $x \to 4^+$ (from the right), the denominator $x - 4$ becomes positive but very small, while the numerator $x + 3$ approaches 7. Thus, the overall fraction tends to positive infinity: $\lim_{x \to 4^+} \frac{x + 3}{x - 4} = +\infty$

Problem 2b:

You are asked to find: $\lim_{x \to 4^-} \frac{x + 3}{x - 4}$

Solution:

As $x \to 4^-$ (from the left), the denominator $x - 4$ becomes negative but very small, while the numerator $x + 3$ still approaches 7. Thus, the overall fraction tends to negative infinity: $\lim_{x \to 4^-} \frac{x + 3}{x - 4} = -\infty$

Summary of Results:

1. $\lim_{x \to 4} \frac{x + 3}{x - 4} = \infty$ (not defined, but depending on the direction, it's $+\infty$ or $-\infty$). 2a. $\lim_{x \to 4^+} \frac{x + 3}{x - 4} = +\infty$ 2b. $\lim_{x \to 4^-} \frac{x + 3}{x - 4} = -\infty$

Would you like more details on any part of the solution?

Here are five related questions you might find useful:

1. What happens if the numerator also approaches zero?
2. Can limits at infinity be applied in other forms of equations?
3. How would this limit change if we added a constant to the denominator?
4. What is the difference between one-sided and two-sided limits?
5. How do we handle indeterminate forms when calculating limits?

Tip: Always check for one-sided limits when the denominator approaches zero, as the sign of the result can change depending on the direction of approach!