Final answer:
The limit of f(x)/g(x) as x approaches infinity is 0. The function ‘f’ approaches a constant, and the derivative of ‘g’ approaches 4. L'Hôpital's rule is applied to determine the limit.
Step-by-step explanation:
The student is asking about the limit of the quotient of two functions ‘f’ and ‘g’ as x approaches infinity.
The given function ‘f’ is f(z) = 32 + 2e⁻³ˣ, which approaches 32 as x approaches infinity because the exponential term approaches zero.
The function ‘g’ has a derivative g'(x) = 4 + 1/x and a limit of infinity as x approaches infinity.
To find the limit of f(x)/g(x) as x approaches infinity, we apply L'Hôpital's rule which states that if the limits of f(x) and g(x) are both zero or both infinity and they are differentiable near a point, then limₓ→∞ f(x)/g(x) can be found by calculating limₓ→∞ f'(x)/g'(x).
Given that as x approaches infinity, e⁻³ˣ approaches zero, the derivative of ‘f’ with respect to x is f'(x) = 0, and the derivative of ‘g’ is g'(x) = 4 + 1/x which approaches 4 as x approaches infinity.
Thus, the limit of f'(x)/g'(x) as x approaches infinity is 0/4 = 0.
Therefore, the limit of f(x)/g(x) as x approaches infinity is also 0.