SOLUTION: Without using l’Hopital’s Rule or Series find Limit(((1+x)^(1/x)-e)/x) as x=0
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Question 1117867
:
Without using l’Hopital’s Rule or Series find
Limit(((1+x)^(1/x)-e)/x) as x=0
Answer by
math_helper(2461)
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Approaching 0 from the negative side:
perl -e '$x=-.1; $e=2.718281828459; for($j=1; $j<6; $j++) { $a = ((1+$x)**(1/$x)-$e)/$x; print "f($x) = $a\n"; $x=$x/10;}'
f(-0.1) = -1.49690162333441
f(-0.01) = -1.37171979700281
f(-0.001) = -1.36038798385263
f(-0.0001) = -1.35926551149801
f(-1e-05) = -1.35915214047877
And approaching 0 from the positive side:
perl -e '$x=.1; $e=2.718281828459; for($j=1; $j<6; $j++) { $a = ((1+$x)**(1/$x)-$e)/$x; print "f($x) = $a\n"; $x=$x/10;}'
f(0.1) = -1.24539368358998
f(0.01) = -1.34679990374718
f(0.001) = -1.35789622340665
f(0.0001) = -1.35901634074731
f(1e-05) = -1.35912667031945
Which looks to be
