Let's evaluate the given limits one by one:

Problem #6 (a):

$\lim_{{x \to 4^+}} \frac{x + 6}{x - 4}$

This is a one-sided limit, where $x$ approaches 4 from the right ($x \to 4^+$).

• The numerator is $x + 6$. As $x \to 4$, this becomes $4 + 6 = 10$.
• The denominator is $x - 4$. As $x \to 4^+$, the denominator approaches $0^+$ (a small positive number).

Thus, the overall limit behaves as:

$\lim_{{x \to 4^+}} \frac{10}{x - 4} = \frac{10}{0^+} = +\infty$

So, the answer to (a) is $+\infty$.

Problem #6 (b):

$\lim_{{x \to 6^-}} \frac{x + 6}{x - 4}$

Here, $x \to 6^-$, which means $x$ approaches 6 from the left.

• The numerator is $x + 6$. As $x \to 6$, this becomes $6 + 6 = 12$.
• The denominator is $x - 4$. As $x \to 6^-$, this becomes $6 - 4 = 2$.

Thus, the overall limit is:

$\lim_{{x \to 6^-}} \frac{12}{2} = 6$

So, the answer to (b) is $6$.

Problem #6 (c):

$\lim_{{x \to 4^-}} \frac{x + 6}{x - 4}$

This is another one-sided limit, where $x$ approaches 4 from the left ($x \to 4^-$).

• The numerator is $x + 6$. As $x \to 4$, this becomes $4 + 6 = 10$.
• The denominator is $x - 4$. As $x \to 4^-$, the denominator approaches $0^-$ (a small negative number).

Thus, the overall limit behaves as:

$\lim_{{x \to 4^-}} \frac{10}{x - 4} = \frac{10}{0^-} = -\infty$

So, the answer to (c) is $-\infty$.

Summary:

(a) $+\infty$

(b) $6$

(c) $-\infty$

Do you have any questions or need further clarification on any part? Here are 5 related questions for you:

1. What is the significance of the one-sided limits $x \to 4^+$ and $x \to 4^-$?
2. Can you explain why the limit in part (b) is finite while the others tend to infinity?
3. How would the limit change if the numerator and denominator were swapped?
4. Can limits approaching infinity occur in real-world applications? Give an example.
5. What is the difference between limits at infinity and infinite limits?

Tip: When evaluating one-sided limits, focus on whether the denominator approaches zero from the positive or negative direction, as this greatly influences the result.