Question

f(3) = 3, \lim\limits_{x \to 3} f(x) = 2; g(3) = 8, \lim\limits_{x \to 3} g(x) = 8; h(3) = 4, \lim\limits_{x \to 3} h(x) = 2. the table above gives selected values and limits of the functions f, g, and h. what is \lim\limits_{x \to 3} \left( h(x)\left(2f(x) + 3g(x)\
ight) \
ight) ? \
options: \
a) 120 \
b) 104 \
c) 56 \
d) 32

Explanation:

Step1: Recall Limit Laws

We use the product rule and sum rule for limits. The limit of a product is the product of the limits, and the limit of a sum is the sum of the limits. So, $\lim_{x\to 3} [h(x)(2f(x) + 3g(x))] = \lim_{x\to 3} h(x) \cdot \lim_{x\to 3} (2f(x) + 3g(x))$.

Step2: Apply Sum and Constant Multiple Rules

For the sum inside, $\lim_{x\to 3} (2f(x) + 3g(x)) = 2\lim_{x\to 3} f(x) + 3\lim_{x\to 3} g(x)$.

Step3: Substitute Known Limits

From the table, $\lim_{x\to 3} h(x) = 2$, $\lim_{x\to 3} f(x) = 2$, and $\lim_{x\to 3} g(x) = 8$. Substitute these values:
First, calculate $2\lim_{x\to 3} f(x) + 3\lim_{x\to 3} g(x) = 2(2) + 3(8) = 4 + 24 = 28$.
Then, multiply by $\lim_{x\to 3} h(x)$: $2 \cdot 28 = 56$? Wait, no, wait. Wait, $\lim_{x\to 3} h(x)$ is 2? Wait, no, the table says $\lim_{x\to 3} h(x) = 2$? Wait, no, let me check again. Wait, the table: $h(3)=4$, $\lim_{x\to 3} h(x)=2$? Wait, no, maybe I misread. Wait, the table:

$f(3)=3$, $\lim_{x\to 3} f(x)=2$

$g(3)=8$, $\lim_{x\to 3} g(x)=8$

$h(3)=4$, $\lim_{x\to 3} h(x)=2$

Wait, no, wait, the problem is $\lim_{x\to 3} [h(x)(2f(x) + 3g(x))]$. So we use the limit of h(x) as x approaches 3, which is 2, limit of f(x) as x approaches 3 is 2, limit of g(x) as x approaches 3 is 8.

So step by step:

1. $\lim_{x\to 3} (2f(x) + 3g(x)) = 2\lim_{x\to 3} f(x) + 3\lim_{x\to 3} g(x) = 2(2) + 3(8) = 4 + 24 = 28$

1. $\lim_{x\to 3} h(x) = 2$

1. Then, the product: $2 \times 28 = 56$? Wait, but option C is 56. Wait, but let me check again. Wait, maybe I made a mistake. Wait, no, the limit of h(x) as x approaches 3 is 2, limit of f(x) is 2, limit of g(x) is 8. So 2(22 + 38) = 2(4 + 24) = 2*28 = 56. So the answer is C.

Answer:

C. 56