https://file.aiquickdraw.com/mathbot/file/1726716649937ssvjhfmo.jpg

Let's go through each of the given limit problems one by one using the provided information.

Given:
\[
\lim_{x \to 2} f(x) = 4, \quad \lim_{x \to 2} g(x) = -2, \quad \lim_{x \to 2} h(x) = 0
\]
We'll compute the limits as follows:

---

**(a) \(\lim_{x \to 2} [f(x) + 5g(x)]\)**

We can split this limit using the sum rule of limits:

\[
\lim_{x \to 2} [f(x) + 5g(x)] = \lim_{x \to 2} f(x) + 5 \cdot \lim_{x \to 2} g(x)
\]

Substitute the given values:

\[
= 4 + 5(-2) = 4 - 10 = -6
\]

So, the limit is **\(-6\)**.

---

**(b) \(\lim_{x \to 2} [g(x)]^3\)**

By the power rule of limits:

\[
\lim_{x \to 2} [g(x)]^3 = \left(\lim_{x \to 2} g(x)\right)^3
\]

Substitute the given value for \(g(x)\):

\[
= (-2)^3 = -8
\]

So, the limit is **\(-8\)**.

---

**(c) \(\lim_{x \to 2} \sqrt{f(x)}\)**

By the square root rule of limits:

\[
\lim_{x \to 2} \sqrt{f(x)} = \sqrt{\lim_{x \to 2} f(x)}
\]

Substitute the given value for \(f(x)\):

\[
= \sqrt{4} = 2
\]

So, the limit is **2**.

---

**(d) \(\lim_{x \to 2} \frac{3f(x)}{g(x)}\)**

Using the quotient rule of limits:

\[
\lim_{x \to 2} \frac{3f(x)}{g(x)} = \frac{3 \cdot \lim_{x \to 2} f(x)}{\lim_{x \to 2} g(x)}
\]

Substitute the given values for \(f(x)\) and \(g(x)\):

\[
= \frac{3 \cdot 4}{-2} = \frac{12}{-2} = -6
\]

So, the limit is **\(-6\)**.

---

**(e) \(\lim_{x \to 2} \frac{g(x)}{h(x)}\)**

Using the quotient rule of limits:

\[
\lim_{x \to 2} \frac{g(x)}{h(x)} = \frac{\lim_{x \to 2} g(x)}{\lim_{x \to 2} h(x)}
\]

Substitute the given values:

\[
= \frac{-2}{0}
\]

This expression results in a division by zero, which means the limit **does not exist**. More specifically, the limit approaches **\(-\infty\)** or **\(\infty\)** depending on the direction of approach, but it is undefined overall.

---

**(f) \(\lim_{x \to 2} \frac{g(x) h(x)}{f(x)}\)**

Using the quotient and product rules of limits:

\[
\lim_{x \to 2} \frac{g(x) h(x)}{f(x)} = \frac{\left(\lim_{x \to 2} g(x)\right) \cdot \left(\lim_{x \to 2} h(x)\right)}{\lim_{x \to 2} f(x)}
\]

Substitute the given values for \(g(x)\), \(h(x)\), and \(f(x)\):

\[
= \frac{(-2)(0)}{4} = \frac{0}{4} = 0
\]

So, the limit is **0**.

---

### Summary of Results:
- (a) \(-6\)
- (b) \(-8\)
- (c) \(2\)
- (d) \(-6\)
- (e) **Does not exist** (division by zero)
- (f) \(0\)

Would you like any additional details or further explanations on any part?

---

Here are 5 related questions for practice:
1. What happens if \( \lim_{x \to 2} g(x) = 0 \) in problem (d)?
2. How would you solve \( \lim_{x \to a} f(x)g(x) \) if \( \lim_{x \to a} g(x) = 0 \)?
3. Can you explain the difference between removable and non-removable discontinuities?
4. How does the limit definition change if \( g(x) \to \infty \) as \( x \to 2 \)?
5. What are the conditions for a limit to exist when dealing with quotient functions?

**Tip:** Always check for the possibility of dividing by zero when dealing with limit problems involving quotient functions.

Given that lim(x->2) f(x) = 4, lim(x->2) g(x) = -2, lim(x->2) h(x) = 0, find the limits for the following expressions: (a) lim(x->2) [f(x) + 5g(x)] (b) lim(x->2) [g(x)]^3 (c) lim(x->2) sqrt(f(x)) (d) lim(x->2) 3f(x) / g(x) (e) lim(x->2) g(x) / h(x) (f) lim(x->2) g(x)h(x) / f(x)

Learn how to compute limits of functions using sum, quotient, and product rules. Explore step-by-step solutions to problems involving f(x), g(x), and h(x) with given limit values. This is an essential skill in high school calculus, suitable for Grades 10-12.

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