2. [-/6 Points] DETAILS MY NOTES SCALCET9 2.3.001. Given that \[ \lim _{x \rightarrow 2} f(x)=4 \quad \lim _{x \rightarrow 2} g(x)=-4 \quad \lim _{x \rightarrow 2} h(x)=0 \] find each limit, if it exists. (If an answer does not exist, enter DNE.) (a) \( \lim _{x \rightarrow 2}[f(x)+3 g(x)] \) \( \square \) (b) \( \lim _{x \rightarrow 2}[g(x)]^{3} \) \( \square \) (c) \( \lim _{x \rightarrow 2} \sqrt{f(x)} \) \( \square \) (d) \( \lim _{x \rightarrow 2} \frac{2 f(x)}{g(x)} \) \( \square \) (e) \( \lim _{x \rightarrow 2} \frac{g(x)}{h(x)} \) \( \square \) (f) \( \lim _{x \rightarrow 2} \frac{g(x) h(x)}{f(x)} \)

Let's evaluate each of the limits step-by-step: (a) \( \lim _{x \rightarrow 2}[f(x)+3 g(x)] \) Using the limit properties, we can split the limit: \[ \lim_{x \to 2} (f(x) + 3g(x)) = \lim_{x \to 2} f(x) + 3 \cdot \lim_{x \to 2} g(x) = 4 + 3 \cdot (-4) = 4 - 12 = -8 \] So, the answer is \(-8\). (b) \( \lim _{x \rightarrow 2}[g(x)]^{3} \) Using the property of limits for powers: \[ \lim_{x \to 2} (g(x))^3 = \left(\lim_{x \to 2} g(x)\right)^3 = (-4)^3 = -64 \] So, the answer is \(-64\). (c) \( \lim _{x \rightarrow 2} \sqrt{f(x)} \) As \(f(x)\) approaches \(4\), we take the square root of that limit: \[ \lim_{x \to 2} \sqrt{f(x)} = \sqrt{\lim_{x \to 2} f(x)} = \sqrt{4} = 2 \] So, the answer is \(2\). (d) \( \lim _{x \rightarrow 2} \frac{2 f(x)}{g(x)} \) Using limit properties: \[ \lim_{x \to 2} \frac{2 f(x)}{g(x)} = \frac{2 \cdot \lim_{x \to 2} f(x)}{\lim_{x \to 2} g(x)} = \frac{2 \cdot 4}{-4} = \frac{8}{-4} = -2 \] So, the answer is \(-2\). (e) \( \lim _{x \rightarrow 2} \frac{g(x)}{h(x)} \) Since \(g(x) \to -4\) and \(h(x) \to 0\), this creates an undefined situation as the limit approaches a constant divided by zero. Thus, we conclude: The limit does not exist (DNE). So, the answer is DNE. (f) \( \lim _{x \rightarrow 2} \frac{g(x) h(x)}{f(x)} \) Here, since \(h(x) \to 0\): \[ \lim_{x \to 2} \frac{g(x) \cdot h(x)}{f(x)} = \frac{\lim_{x \to 2} g(x) \cdot \lim_{x \to 2} h(x)}{\lim_{x \to 2} f(x)} = \frac{-4 \cdot 0}{4} = \frac{0}{4} = 0 \] So, the answer is \(0\). Results Summary: (a) -8 (b) -64 (c) 2 (d) -2 (e) DNE (f) 0