Question

CALCULUS: Which of the following functions grows the fastest as x goes to infinity?

Answer 1

Answer:

It is A. 2^x

Step-by-step explanation:

For relative rates of growth, we know: Log < Poly < Exponential.

This rules out B, C, and D quickly. Although, if you do L'Hôpital's rule, it might seem like A & D are very comparable. However, if you continually took the derivative of 2^x and x^20, eventually x^20 would equal zero, while 2^x would still be a variable. So, you would have:

$\lim_{x \to \infty} \frac{2^{x}}{0} =$ ∞

When the limit of x as it approaches infinity is infinity, then f(x) grows at a faster rate than g(x).

Also, by looking at the graph (which is attached), you can see that 2^x grows faster than x^20.