Question

1. (2pts) for each pair of functions, circle the function that dominates as ( x \to infty ).

a. ( 10e^{0.1x} ) or ( 5000x^2 )
b. ( 2x^4 ) or ( 10x^3 + 25x^2 + 50x + 100 )

Explanation:

Step1: Analyze exponential vs polynomial

Recall that exponential functions grow faster than any polynomial function as $x \to \infty$. For $10e^{0.1x}$ (exponential) and $5000x^2$ (polynomial), the exponential dominates.

Step2: Analyze higher vs lower degree polynomial

For polynomials, the function with the highest degree dominates as $x \to \infty$. $2x^4$ has degree 4, while $10x^3 + 25x^2 + 50x + 100$ has degree 3, so the degree 4 polynomial dominates.

Answer:

a. $10e^{0.1x}$
b. $2x^4$