25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,
dy/dt = -0.6y
when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

41. lim (x → 0⁺) (ln x)² / ln(sin x)

Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.

9. (sinh(x)+cosh(x))⁴

130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = x³ + 1

Verify the integration formulas in Exercises 37–40.

39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C

Evaluate the integrals in Exercises 53–76.

57. ∫dx/(x√(25x²-2))