If `f(x)=(1)/((x+1)!)`, what is the smallest integer x such that `f(x) lt 0.000005` ?

Options

A. 7
B. 8
C. 9
D. 10

Answer

B

Explanation

First convert `0.000005` into a fraction :
`0.000005=(5)/(1,000,000)=(1)/(200,000)`
You're looking for the smallest integer value of x that will make `(1)/((x+1)!)` less than `(1)/(200,000)`.
In other words, you're looking for the smallest integer x that will make `(x+1)!` greater than 200,000. Use your calculator and try a few possibilities. `1xx2xx3xx4xx5xx6xx7xx8=40,320`. Not big enough. But multiply that by 9 and you're up to 362,880. So `9! gt 200,000, x + 1 = 9`, and therefore x = 8.