SOLUTION: Without using l’Hopital’s Rule or Series find Limit(tan(x))^(2x-pi) as x=pi/2

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Question 1117866: Without using l’Hopital’s Rule or Series find
Limit(tan(x))^(2x-pi) as x=pi/2
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
When I try this using Perl, I get a limit of 1 when approaching x+=+pi+%2F+2+ from the left.
But approaching +x=+pi%2F2+ from the right, there seems to be a discontinuity. Based on this data, I'd conclude the limit does not exist (the limits Lim { f(x=a+) } and Lim{ f(x=a-) } must match to say Lim{f(x=a) } exists).

From the left:
perl -e 'use Math::Trig; $pi=3.14159265359; $x=(31/64)*$pi; for($j=0; $j<10; $j++) { $a = tan($x)**(2*$x-$pi); $t=$x/$pi; print "f($t * pi) = $a\n"; $x=($x+($pi/2))/2;}'
f(0.484375 * pi) = 0.743910236876359
f(0.4921875 * pi) = 0.833624828514047
f(0.49609375 * pi) = 0.897625801099151
f(0.498046875 * pi) = 0.939405869042598
f(0.4990234375 * pi) = 0.965115993386296
f(0.49951171875 * pi) = 0.980316256788529
f(0.499755859375 * pi) = 0.989057015331787
f(0.4998779296875 * pi) = 0.993984876614831
f(0.49993896484375 * pi) = 0.99672291920332
f(0.499969482421875 * pi) = 0.998227432454653


From the right:
perl -e 'use Math::Trig; $pi=3.14159265359; $x=(33/64)*$pi; for($j=0; $j<10; $j++) { $a = tan($x)**(2*$x-$pi); $t=$x/$pi; print "f($t * pi) = $a\n"; $x=($x+($pi/2))/2;}'
f(0.515625 * pi) = NaN (NaN = Not a Number)
f(0.5078125 * pi) = NaN
f(0.50390625 * pi) = NaN
f(0.501953125 * pi) = NaN
f(0.5009765625 * pi) = NaN
f(0.50048828125 * pi) = NaN
f(0.500244140625 * pi) = NaN
f(0.5001220703125 * pi) = NaN
f(0.50006103515625 * pi) = NaN
f(0.500030517578125 * pi) = NaN