Evaluate the limit without L'Hospital's Rule and Taylor Series: limx→1(x−1)tan(πx2)
I can't seem to go find a substitution that works for this limit. I tried with u=x−1, but do not know where to go from there.
Answer
limx→1(x−1)tan(πx2)→[t=x−1t→0]→limt→0t⋅tan(πt2+π2)=−limt→0t⋅cot(πt2)==−limt→0tcos(πt2)sin(πt2)=−limt→0cos(πt2)sin(πt2)⋅π2tπ2=−limt→02cos(πt2)π⋅1=−2π.
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