Final answer:
Graphically, the curve of f(x) approaches the horizontal line of g(x) = e^3 as x increases. Numerically, by evaluating f(x) for large values of x and comparing it to e^3, we see that the values converge to approximately 20.086.
Step-by-step explanation:
To show that the value of f(x) = (1 + 3/x)^x approaches the value of g(x) = e^3 as x increases without bound graphically and numerically, we need to consider the behavior of f(x) as x grows large.
Graphically
On a graph, f(x) will be represented by a curve. As x increases, the term 3/x becomes insignificant compared to 1, which means (1 + 3/x) approaches 1. According to the properties of exponents, as x gets larger, f(x) should approach e^3, which is the constant value of g(x). Plotting these functions on a coordinate plane, you would see that the curve of f(x) gets closer to the horizontal line y = e^3, which is approximately 20.086.
Numerically
Numerically, we can show this by evaluating f(x) for increasing values of x and observing that the results get closer to 20.086. For instance, for large values of x, such as x=100 or x=1000, the value of f(x) will round to three decimal places much closer to e^3, reflecting that f(x) is approaching g(x) as x becomes large.