Which of the following functions grows the fastest as x goes to infinity?
3^x
ln(x)
e^4x
x^10
I put e^4x but I thought that 3^x might have been it too. I know for sure that it is not ln(x) and x^10 because those grow much slower.
yes, e^(4x) grows at the fastest rate
e.g. let x = 20, (not very large at all)
3^20 = 3.48 x 10^9
ln(20) = only about 3
e^(4x) = 5.5 x 10^34 <------
20^10 = 1.024 x 10^13
To determine which function grows the fastest as x goes to infinity, we need to compare their growth rates. One way to do this is by examining the limits of these functions as x becomes infinitely large.
Let's analyze each function separately:
1. 3^x:
To see if 3^x grows faster than e^4x, we can take the limit of their ratio as x approaches infinity:
lim(x → ∞) (3^x / e^4x)
Applying L'Hôpital's rule by differentiating the numerator and denominator with respect to x, we have:
lim(x → ∞) (ln(3) * 3^x / (4 * e^4x))
Since the derivative in the numerator is a constant multiple of the original function, 3^x, and the derivative in the denominator is also a constant multiple of e^4x, the limit remains unchanged:
lim(x → ∞) (ln(3) * 3^x / (4 * e^4x)) = ln(3) / 4
So, the limit is a finite value, indicating that 3^x and e^4x grow at the same rate as x goes to infinity.
2. ln(x):
The natural logarithm function, ln(x), grows much slower than exponentials as x goes to infinity. Therefore, it is not the function that grows the fastest.
3. e^4x:
We will compare this function with the previous one, 3^x. Taking the limit of their ratio:
lim(x → ∞) (e^4x / 3^x)
Using L'Hôpital's rule by differentiating both the numerator and denominator:
lim(x → ∞) (4e^4x / (ln(3) * 3^x))
Taking only the exponential functions into consideration:
lim(x → ∞) (4e^4x / 3^x) = ∞
The limit evaluates to infinity, indicating that e^4x grows faster than 3^x as x approaches infinity.
4. x^10:
The polynomial function, x^10, also grows slower than exponential functions. Thus, it is not the function that grows the fastest.
In conclusion, among the given functions, e^4x grows the fastest as x goes to infinity.
To determine which function grows the fastest as x goes to infinity, we can compare the rates of growth of the given functions. We will compare them using their respective limits as x approaches infinity.
1. Limit as x approaches infinity of 3^x:
Taking the limit of 3^x as x approaches infinity, we get ∞. Therefore, 3^x grows exponentially and very fast as x goes to infinity.
2. Limit as x approaches infinity of ln(x):
Taking the limit of ln(x) as x approaches infinity, we get ∞. However, ln(x) grows much slower than exponential functions like 3^x.
3. Limit as x approaches infinity of e^4x:
Taking the limit of e^4x as x approaches infinity, we get ∞. Similar to 3^x, e^4x is also an exponential function and grows very fast as x goes to infinity.
4. Limit as x approaches infinity of x^10:
Taking the limit of x^10 as x approaches infinity, we get ∞. However, x^10 grows slower than the exponential functions above.
Based on the comparison, we can conclude that both 3^x and e^4x grow the fastest as x goes to infinity. Both functions have exponential growth, but 3^x grows at a faster rate when compared to e^4x. Therefore, 3^x is the correct answer.