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EXAMPLE 4Using L’Hôpital’s Rule to Find a Limit at Infinity

Find:

  1. (a) limxlnxx
  2. (b) limxxex
  3. (c) limxexx

Solution (a) Since limxlnx= and limxx=, lnxx is an indeterminate form at of the type . Using L’Hôpital’s Rule, we have limxlnxx=limxddxlnxddxx=limx1x1=limx1x=0L'Hˆopital'sRule

(b) limxx= and limxex=, so xex is an indeterminate form at of the type . Using L’Hôpital’s Rule, we have limxxex=limxddxxddxex=limx1ex=0L'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 limxexx=limxddxexddxx=limxex1=L'Hˆopital'sRule