Barrons AP Calculus - David Bock

(Note the easier solution

Other indeterminate forms, such as 0^0 , 1∞ and ∞^0 , may be resolved by taking the natural logarithm
and then applying L’Hôpital’s Rule.

BC ONLY

EXAMPLE 46

Find

SOLUTION: is of type 1∞. Let y = (1 + x)1/x, so that

ln ln (1 + x). Then ln which is of type Thus,

and since ln y = 1, y = e^1 = e.

EXAMPLE 47

Find

SOLUTION: is of type ∞^0. Let y = x1/x, so that ln

(which, as x → ∞, is of type ). Then ln and y = e^0 = 1.

For more practice, redo the Practice Exercises, applying L’Hôpital’s Rule wherever possible.

K. RECOGNIZING A GIVEN LIMIT AS A DERIVATIVE

It is often extremely useful to evaluate a limit by recognizing that it is merely an expression for the
definition of the derivative of a specific function (often at a specific point). The relevant definition is
the limit of the difference quotient:

EXAMPLE 48

Find