Final answer:
To find the limit of f(x)/g(x) as x approaches infinity, we analyze the behavior of f(x) as roughly linear in 3x and g(x) as also linear due to its derivative. With both functions growing linearly, the ratio approaches the constant 3/4 as x goes towards infinity. So, the correct answer is b. 3/4.
Step-by-step explanation:
The question asks to find the value of the limit limₙᵢ→∞ f(x)/g(x), where the function f(x) is defined as f(x)=3x+2e⁻³ˣ, and g is a differentiable function with a derivative g'(x)=4+ 1/x and limₙᵢ→∞ g(x) = ∞. To solve this, we want to look at the behavior of each function as x approaches infinity.
Firstly, consider the function f(x). As x goes to infinity, the term 3x will dominate because the exponential term 2e⁻³ˣ will approach 0. Therefore, f(x) roughly behaves like 3x as x goes to infinity.
Now, consider g(x). Since g'(x) = 4 + 1/x, we can infer that g(x) is increasing and because the limit as x approaches infinity of g(x) is infinity, we can deduce that g(x) is dominantly behaving like a linear function as x goes to infinity.
Dividing f(x) by g(x), the 3x term in f(x) divided by the linearly increasing term in g(x) will give us a ratio that approaches a constant value. Since there is no indication from the question to suggest otherwise, and both f(x) and g(x) grow linearly without bounds, we might expect the limit of the ratio to exist.
Given that g(x) increases without bound linearly, the coefficient of x in g(x) will determine the behavior of the quotient. As g'(x) approaches 4 as x approaches infinity, we can assume the leading term to be 4x (since the 1/x term in g'(x) goes to 0). So the limit can be inferred as the ratio of coefficients of x, which is 3/4.
Therefore, the option for the correct answer is b. 3/4.