Which of the following functions grows the fastest as x goes to infinity?
a) f(x) √x
b) f(x) √(x+1)
c) f(x) √(10x+1)
d) They all grow at the same rate as x goes to infinity
for large x,
√(x+1) ≈ √x
√(10x+1) ≈ √10 √x
so, what do you think?
answer C?
Yes , The square root of 10 is greater than one.
No its not C
Well, if we take a closer look at these options, we can start eliminating some choices.
Option a) f(x) √x, well, the square root of x grows slower than the square root of (x + 1) or the square root of (10x + 1), so we can rule this one out.
Now let's compare options b) f(x) √(x + 1) and c) f(x) √(10x + 1). As x goes to infinity, the 10x + 1 term in option c) dominates the x + 1 term in option b). Thus, option c) will grow faster.
So, the fastest-growing function as x goes to infinity is option c) f(x) √(10x + 1).
But hey, at least they're all still growing, unlike my hopes and dreams at the moment.
To determine which function grows the fastest as x goes to infinity, we can compare the growth rates by taking the limit as x approaches infinity for each function.
a) f(x) √x: We take the limit as x approaches infinity:
lim(x->∞) √x = ∞
b) f(x) √(x+1): We take the limit as x approaches infinity:
lim(x->∞) √(x+1) = ∞
c) f(x) √(10x+1): We take the limit as x approaches infinity:
lim(x->∞) √(10x+1) = ∞
From the above calculations, we can see that the limit is ∞ for all the functions as x goes to infinity. Therefore, the functions a), b), and c) all grow at the same rate as x goes to infinity.
The answer is option d) They all grow at the same rate as x goes to infinity.