Both of these functions grow as x gets larger and larger. Which function eventually exceeds the other?
To determine which function eventually exceeds the other as x gets larger and larger, we need to compare their growth rates.
Let's look at the functions f(x) = 2^x and g(x) = x^2.
As x approaches infinity, the exponential function 2^x grows much faster than the polynomial function x^2. This can be seen by comparing the growth rates of the two functions:
lim (x->∞) 2^x = ∞ (exponential growth)
lim (x->∞) x^2 = ∞ (polynomial growth)
Therefore, the exponential function f(x) = 2^x will eventually exceed the polynomial function g(x) = x^2 as x gets larger and larger.