Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Let's solve these limit problems step-by-step:
Given:
[tex]\[ \lim_{x \to 3} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3} g(x) = -7 \][/tex]
### Part (a)
[tex]\[ \lim_{x \to 3} [f(x) g(x)] \][/tex]
Using the limit property that states [tex]\(\lim_{x \to a} [f(x) \cdot g(x)] = \left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to a} g(x)\right)\)[/tex], we have:
[tex]\[ \lim_{x \to 3} [f(x) g(x)] = \left(\lim_{x \to 3} f(x)\right) \cdot \left(\lim_{x \to 3} g(x)\right) = 4 \cdot (-7) = -28 \][/tex]
So,
[tex]\[ \lim_{x \to 3} [f(x) g(x)] = -28 \][/tex]
### Part (b)
[tex]\[ \lim_{x \to 3} [3 f(x) g(x)] \][/tex]
We can factor out the constant 3 from the limit:
[tex]\[ \lim_{x \to 3} [3 f(x) g(x)] = 3 \cdot \lim_{x \to 3} [f(x) g(x)] \][/tex]
From part (a), we know:
[tex]\[ \lim_{x \to 3} [f(x) g(x)] = -28 \][/tex]
So,
[tex]\[ \lim_{x \to 3} [3 f(x) g(x)] = 3 \cdot (-28) = -84 \][/tex]
### Part (c)
[tex]\[ \lim_{x \to 3} [f(x) + 8 g(x)] \][/tex]
Using the limit property that states [tex]\(\lim_{x \to a} [f(x) + g(x)] = \left(\lim_{x \to a} f(x)\right) + \left(\lim_{x \to a} g(x)\right)\)[/tex], we have:
[tex]\[ \lim_{x \to 3} [f(x) + 8 g(x)] = \lim_{x \to 3} f(x) + 8 \cdot \lim_{x \to 3} g(x) \][/tex]
Given,
[tex]\[ \lim_{x \to 3} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3} g(x) = -7 \][/tex]
So,
[tex]\[ \lim_{x \to 3} [f(x) + 8 g(x)] = 4 + 8 \cdot (-7) = 4 + (-56) = 4 - 56 = -52 \][/tex]
### Part (d)
[tex]\[ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] \][/tex]
Using the limit properties and given:
[tex]\[ \lim_{x \to 3} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3} g(x) = -7 \][/tex]
First, we find the limit of the denominator:
[tex]\[ \lim_{x \to 3} [f(x) - g(x)] = \lim_{x \to 3} f(x) - \lim_{x \to 3} g(x) = 4 - (-7) = 4 + 7 = 11 \][/tex]
Since the denominator does not approach zero, we can use the limit properties for division:
[tex]\[ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] = \frac{\lim_{x \to 3} f(x)}{\lim_{x \to 3} [f(x) - g(x)]} = \frac{4}{11} \][/tex]
So, we have:
[tex]\[ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] = \frac{4}{11} \approx 0.36363636363636365 \][/tex]
### Summary
[tex]\[ \begin{aligned} \lim_{x \to 3} [f(x) g(x)] &= -28, \\ \lim_{x \to 3} [3 f(x) g(x)] &= -84, \\ \lim_{x \to 3} [f(x) + 8 g(x)] &= -52, \\ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] &\approx 0.36363636363636365. \end{aligned} \][/tex]
Thus, the answers are:
a. [tex]\(-28\)[/tex]
b. [tex]\(-84\)[/tex]
c. [tex]\(-52\)[/tex]
d. [tex]\(\approx 0.3636\)[/tex]
Given:
[tex]\[ \lim_{x \to 3} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3} g(x) = -7 \][/tex]
### Part (a)
[tex]\[ \lim_{x \to 3} [f(x) g(x)] \][/tex]
Using the limit property that states [tex]\(\lim_{x \to a} [f(x) \cdot g(x)] = \left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to a} g(x)\right)\)[/tex], we have:
[tex]\[ \lim_{x \to 3} [f(x) g(x)] = \left(\lim_{x \to 3} f(x)\right) \cdot \left(\lim_{x \to 3} g(x)\right) = 4 \cdot (-7) = -28 \][/tex]
So,
[tex]\[ \lim_{x \to 3} [f(x) g(x)] = -28 \][/tex]
### Part (b)
[tex]\[ \lim_{x \to 3} [3 f(x) g(x)] \][/tex]
We can factor out the constant 3 from the limit:
[tex]\[ \lim_{x \to 3} [3 f(x) g(x)] = 3 \cdot \lim_{x \to 3} [f(x) g(x)] \][/tex]
From part (a), we know:
[tex]\[ \lim_{x \to 3} [f(x) g(x)] = -28 \][/tex]
So,
[tex]\[ \lim_{x \to 3} [3 f(x) g(x)] = 3 \cdot (-28) = -84 \][/tex]
### Part (c)
[tex]\[ \lim_{x \to 3} [f(x) + 8 g(x)] \][/tex]
Using the limit property that states [tex]\(\lim_{x \to a} [f(x) + g(x)] = \left(\lim_{x \to a} f(x)\right) + \left(\lim_{x \to a} g(x)\right)\)[/tex], we have:
[tex]\[ \lim_{x \to 3} [f(x) + 8 g(x)] = \lim_{x \to 3} f(x) + 8 \cdot \lim_{x \to 3} g(x) \][/tex]
Given,
[tex]\[ \lim_{x \to 3} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3} g(x) = -7 \][/tex]
So,
[tex]\[ \lim_{x \to 3} [f(x) + 8 g(x)] = 4 + 8 \cdot (-7) = 4 + (-56) = 4 - 56 = -52 \][/tex]
### Part (d)
[tex]\[ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] \][/tex]
Using the limit properties and given:
[tex]\[ \lim_{x \to 3} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3} g(x) = -7 \][/tex]
First, we find the limit of the denominator:
[tex]\[ \lim_{x \to 3} [f(x) - g(x)] = \lim_{x \to 3} f(x) - \lim_{x \to 3} g(x) = 4 - (-7) = 4 + 7 = 11 \][/tex]
Since the denominator does not approach zero, we can use the limit properties for division:
[tex]\[ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] = \frac{\lim_{x \to 3} f(x)}{\lim_{x \to 3} [f(x) - g(x)]} = \frac{4}{11} \][/tex]
So, we have:
[tex]\[ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] = \frac{4}{11} \approx 0.36363636363636365 \][/tex]
### Summary
[tex]\[ \begin{aligned} \lim_{x \to 3} [f(x) g(x)] &= -28, \\ \lim_{x \to 3} [3 f(x) g(x)] &= -84, \\ \lim_{x \to 3} [f(x) + 8 g(x)] &= -52, \\ \lim_{x \to 3} \left[\frac{f(x)}{f(x) - g(x)}\right] &\approx 0.36363636363636365. \end{aligned} \][/tex]
Thus, the answers are:
a. [tex]\(-28\)[/tex]
b. [tex]\(-84\)[/tex]
c. [tex]\(-52\)[/tex]
d. [tex]\(\approx 0.3636\)[/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
