Solve the initial value problems in Exercises 87 and 88.
88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)

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

y = e^(5-7x)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

29. y = (1 - t)coth⁻¹(√t)

For Exercises 127 and 128 find a function f satisfying each equation.

127. ∫₂ˣ √(f(t)) dt = x ln x

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

29. y = ln(1/(x√(x+1)))

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

60. lim (x → 0) (e^x + x)^(1/x)