Find:
Solution (a) Since limx→∞lnx=∞ and limx→∞x=∞, lnxx is an indeterminate form at ∞ of the type ∞∞. Using L’Hôpital’s Rule, we have limx→∞lnxx=limx→∞ddxlnxddxx=limx→∞1x1=limx→∞1x=0↑L'Hˆopital'sRule
(b) limx→∞x=∞ and limx→∞ex=∞, so xex is an indeterminate form at ∞ of the type ∞∞. Using L’Hôpital’s Rule, we have limx→∞xex=limx→∞ddxxddxex=limx→∞1ex=0↑L'Hˆopital'sRule
(c) From (b), we know that exx is an indeterminate form at ∞ of the type ∞∞. Using L’Hô pital’s Rule, we have limx→∞exx=limx→∞ddxexddxx=limx→∞ex1=∞↑L'Hˆopital'sRule