SOLUTION: is of type 1∞. Let y = (1 + x)1/x, so that ln y = ln (1 +
x). Then , which is of type . Thus,
and since
Example 47 __
Find .
SOLUTION: is of type ∞^0 . Let y = x1/x, so that ln (which, as
x → ∞, is of type ). Then , and .
*The limit can be finite or infinite (+∞ or −∞).
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
SOLUTION: is the derivative of f (x) = x 4 at the point where x =
- Since
f ′(x) = 4 x 3 , the value of the given limit is f ′(2) = 4(2^3 ) = 32.