Sunday, November 3, 2019

calculus - Solving the following limit without L'Hospital Rule: limxto1left(x1right)tanleft(fracpix2right)?




Evaluate the limit without L'Hospital's Rule and Taylor Series: limx1(x1)tan(πx2)



I can't seem to go find a substitution that works for this limit. I tried with u=x1, but do not know where to go from there.


Answer



limx1(x1)tan(πx2)[t=x1t0]limt0ttan(πt2+π2)=limt0tcot(πt2)==limt0tcos(πt2)sin(πt2)=limt0cos(πt2)sin(πt2)π2tπ2=limt02cos(πt2)π1=2π.


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