SOLUTION: find the limit as x approaches 1^+ of lnx tan(pi(x)/2) answer: -2/pi how do you solve for the limit?

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Question 374259: find the limit as x approaches 1^+ of
lnx tan(pi(x)/2)
answer: -2/pi
how do you solve for the limit?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

matrix%282%2C1%2Clim%2C%22x-%3E1%22%5E%22%2B%22%29%28ln%28x%29%2ATan%28pi%2Ax%2F2%29%29

Write the tangent expression as the reciprocal of the cotangent

matrix%282%2C1%2Clim%2C%22x-%3E1%22%5E%22%2B%22%29%28ln%28x%29%28expr%281%2FCot%28pi%2Ax%2F2%29%29%29%29
 
which is the same as:

matrix%282%2C1%2Clim%2C%22x-%3E1%22%5E%22%2B%22%29%28expr%28%28ln%28x%29%29%2FCot%28pi%2Ax%2F2%29%29%29%29

Since both numerator and denominator approach 0 as x approaches -1,
we can use L'Hopital's rule by taking deritatives of top and bottom:

[The software available here writes Csc²x as Csc²%22%22%2A%22%22x and same
for other trig functions squared. Sorry! )

matrix%282%2C1%2Clim%2C%22x-%3E1%22%5E%22%2B%22%29%28expr%281%2Fx%29%2F%28-Csc%5E2%28pi%2Ax%2F2%29%2A%28pi%2F2%29%29%29%29%29

Write the cosecant squared in the denominator as a sine squared in the
numerator, and remember to keep the - sign:

matrix%282%2C1%2Clim%2C%22x-%3E1%22%5E%22%2B%22%29%28%28-expr%281%2Fx%29%2ASin%5E2%28pi%2Ax%2F2%29%29%2F%28%28pi%2F2%29%29%29%29%29

Change division by pi%2F2 as multiplication by its reciprocal 2%2Fpi%29

matrix%282%2C1%2Clim%2C%22x-%3E1%22%5E%22%2B%22%29%28-expr%281%2Fx%29%2ASin%5E2%28pi%2Ax%2F2%29%2A%282%2Fpi%29%29%29%29

Now when x is close to 1 this is close to what we get when we
substitute 1 for x

-expr%281%2F%281%29%29%2ASin%5E2%28pi%2A%281%29%2F2%29%2A%282%2Fpi%29%29%29

-Sin%5E2%28pi%2F2%29%2A%282%2Fpi%29%29

-%281%29%5E2%2A%282%2Fpi%29%29

-2%2Fpi

Edwin