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Let's break down the problem step by step. We are given a piecewise function \( h(x) \):

\[
h(x) = 
\begin{cases} 
\frac{3x - 12}{x^2 - 4x} & \text{if } x < 4 \\
\frac{x^2 - 6}{x + 1} & \text{if } x \geq 4
\end{cases}
\]

You need to evaluate the following expressions:

**(a)** \( h(4) \)

Since \( h(x) = \frac{x^2 - 6}{x + 1} \) for \( x \geq 4 \), we can directly substitute \( x = 4 \):

\[
h(4) = \frac{4^2 - 6}{4 + 1} = \frac{16 - 6}{5} = \frac{10}{5} = 2
\]

So, \( h(4) = 2 \).

**(b)** \( \lim_{x \to 4^+} h(x) \)

For \( x \to 4^+ \), we use the part of the function where \( x \geq 4 \), which is \( h(x) = \frac{x^2 - 6}{x + 1} \). Taking the limit as \( x \to 4^+ \):

\[
\lim_{x \to 4^+} h(x) = \frac{4^2 - 6}{4 + 1} = \frac{10}{5} = 2
\]

Thus, \( \lim_{x \to 4^+} h(x) = 2 \).

**(c)** \( \lim_{x \to 4^-} h(x) \)

For \( x \to 4^- \), we use the part of the function where \( x < 4 \), which is \( h(x) = \frac{3x - 12}{x^2 - 4x} \). Factor the numerator and denominator:

\[
h(x) = \frac{3(x - 4)}{x(x - 4)}
\]

For \( x \neq 4 \), we can simplify this expression to:

\[
h(x) = \frac{3}{x}
\]

Taking the limit as \( x \to 4^- \):

\[
\lim_{x \to 4^-} h(x) = \frac{3}{4}
\]

Thus, \( \lim_{x \to 4^-} h(x) = \frac{3}{4} \).

### Final Answers:
(a) \( h(4) = 2 \)  
(b) \( \lim_{x \to 4^+} h(x) = 2 \)  
(c) \( \lim_{x \to 4^-} h(x) = \frac{3}{4} \)

Would you like further details on any of these steps?

### Related Questions:
1. How do you find limits for piecewise functions?
2. Why is there a difference between the limits from the left and right in this example?
3. How would the function behave at \( x = 4 \) if it were continuous?
4. What is the significance of factoring the expression before taking the limit?
5. How does the definition of the piecewise function impact the limits?

### Tip:
When evaluating limits for piecewise functions, always check which piece of the function applies to the left-hand limit and the right-hand limit.

Consider the following piecewise defined function: h(x) = (3x-12)/(x^2-4x) for x < 4, and h(x) = (x^2 - 6)/(x + 1) for x >= 4. Evaluate the following expressions: (a) h(4), (b) lim(x -> 4^+) h(x), (c) lim(x -> 4^-) h(x).

Learn how to evaluate the piecewise function h(x) and compute one-sided limits as x approaches 4. Includes steps for factoring and limit calculation. Suitable for advanced high school students in Grades 11-12.

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