Limits. Determine (exact values) the domain and range of the function

Question

 Limits. Determine (exact values) the domain and range of the function

F(x) = (1+10^x/2+x^2)^1/2x

You may assume that f is increasing on its domain. You must show the details of all limits used to verify your range.

Answer 1

The domain isn't that bad for exponential functions. Essentially, the only x values that won't work are x values that yield 0⁰. There are no x values that do that, so the domain is all real values. For the range, we can utilize the fact that our function is increasing. Thus the lowest value of f occurs when x = -∞ and the highest value of f occurs when x = +∞. We can't just plug in ∞ and -∞. We'll have to use limits

lim f(x) = (∞)^-∞ = 1/(∞^∞) = 0

x→-∞

lim f(x) = ∞^∞ = ∞

x→∞

Thus the range is from 0 to ∞