Question

Consider the function below.

f(x) = (1 - (4/x))ˣ
(a) Use a graph to estimate the value of the limit
x→[infinity] f(x) .correct to two decimal places.
(b) Use a table of values of f(x) to estimate the limit to four decimal places.

Answer 1

The given question involves estimating the limit of the function f(x) = (1 - (4/x))^x as x approaches infinity both graphically and numerically. A graphing utility can be used to visualize the function and estimate the limit, while a table of values can provide a more precise numerical estimation.

Step-by-step explanation:

The student asked a question related to finding the limit of the function f(x) = (1 - (4/x))^x as x approaches infinity. To estimate the limit graphically (part a), one would plot the function on a graphing utility and observe the y-value that the function is approaching as x becomes large. To accurately estimate the graph's limit to two decimal places, the graph should be zoomed in on the y-axis around the value where the function appears to stabilize as x increases.

For part b, creating a table of values of f(x) for large values of x would allow us to estimate the limit to four decimal places more precisely. As x grows, we should see the values of f(x) approach a constant number. This numerical approach involves calculating values of f(x) at various large x values (such as 100, 1000, 10000, etc.) and noting the convergence of f(x) to a specific value.