In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x

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

24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = xe^x-e^x

22. The function ln x grows slower than any polynomial Show that ln(x) grows slower as x→∞ than any nonconstant polynomial.

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

119. y = (sin x)ˣ

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

15. lim(x→∞)arctan(x)

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

108. lim(x→∞)(e^x arctan(e^x))/(e^(2x)+x)