Compare the rates of growth of f(x) = x + sinx and g(x) = x as x approaches infinity.
f(x) grows faster than g(x) as x goes to infinity.
g(x) grows faster than f(x) as x goes to infinity.
f(x) and g(x) grow at the same rate as x goes to infinity.
The rate of growth cannot be determined.
I think it is A because f(x) would grow faster because of the sinx
They both grow at the same rate. D is a trick answer.
f(x) and g(x) grow at the same rate as x goes to infinity is the right answer.
Thanks Joy :D
You are correct! In order to determine the rates of growth of f(x) = x + sinx and g(x) = x as x approaches infinity, we can compare the magnitudes of their derivatives.
The derivative of f(x) with respect to x is f'(x) = 1 + cosx.
The derivative of g(x) with respect to x is g'(x) = 1.
As x approaches infinity, the values of cosx will oscillate between -1 and 1, but it will never exceed these bounds. Thus, the absolute value of cosx remains bounded. Since 1 is a constant, the derivative of f(x) will always remain bounded as x goes to infinity.
On the other hand, the derivative of g(x) is just a constant 1, which does not depend on x. Therefore, it remains constant as x goes to infinity.
Since the derivative of f(x) is bounded while the derivative of g(x) is constant, we can conclude that the rate of growth of f(x) is faster than the rate of growth of g(x) as x approaches infinity. Hence, option A is the correct answer.
geeet dunked on!
disagree,
the value of sinx fluctuates between -1 and 1, so as x approaches infinity, adding or subtracting a number between 0 and 1 will be insignificant.
The values of f(x) and g(x) will differ by at most 1
- their rates of growth would vary just be a bit
My pick would be the last one, cannot be determined.
Take the derivative of each to find the rates of growth.
f'(x) = 1 + cos(x)
g'(x) = 1
lim_{x→∞} g'(x) = 1 but
lim_{x→∞} f'(x) does not exist because cos(x) varies between -1 and 1.
Therefore, the rates of growth cannot be compared.