Help needed with math, please and thank you

Question

Compare the rates of growth of f(x) = x + sinx and g(x) = x as x approaches infinity.

Options:

1)	f(x) grows faster than g(x) as x goes to infinity.
2)	g(x) grows faster than f(x) as x goes to infinity.
3)	f(x) and g(x) grow at the same rate as x goes to infinity.
4)	The rate of growth cannot be determined.

Answer 1

We must ask, does the independent variable, f or g, grow at a faster rate.

If we sub f(10) and g(10), we get:

f(10) = 10 - 0.544 = 9.46

g(10) = 10

when x = 10, g(x) > f(x)

However, if we sub in f(20) and g(20):

f(20) = 20 + .913 = 20.91

g(20) = 20

when x = 20, f(x) > g(x) this time

Thus with values approaching infinity, we cannot determine which growth rate is greater, nor are the rates equal, so d would be correct.