Find the limit of the complex function.limx→a(2−xa)(tan⁡πx2a)I have simplified this limit to this extent :elimx→a((1−xa)(tan⁡πx2a))I don't know how to simplify the limit after that.

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Find the limit of the complex function.limx→a(2−xa)(tan⁡πx2a)I have simplified this limit to this extent :elimx→a((1−xa)(tan⁡πx2a))I don&#039;t know how to simplify the limit after that.

wyjadaczeqa8 · Accepted Answer

log⁡(2−xa)tan⁡(πx2a)=(tan⁡πx2a)(log⁡(2−xa))=sin⁡(πx2a)log⁡(1+(1−xa))cos⁡(π2(xa−1)+π2)=sin⁡(πx2a)(1−xa)−12(1−xa)2+⋯π2(1−xa)−13!(π2(1−xa))3+⋯=sin⁡(πx2a)1−12(1−xa)+⋯π2−13!(π2)3(1−xa)2+⋯taking limit as x→a, then log⁡(2−xa)tan⁡(πx2a)→2π

Kendall Clark · Accepted Answer

From the given limit, using t=x-a, you can rewrite it as:limt→0(1−ta)−cot⁡(πt2a)This can be converted to the form e… using the definition of exponent function, by multiplying and dividing in the power by -t/a. Thus we have:limt→0(1−ta)−cot⁡(πt2a)×−at×−ta=limt→0ecot⁡(πt2a)×taThen due continuity of ex, we have:limt→0ecot⁡(πt2a)×ta=elimt→0cot⁡(πt2a)×ta=exp⁡(limt→0ta×π2×2πtan⁡(πt2a))=exp⁡(2π)

Find the limit of the complex function. \(\displaystyle\lim_{{{x}\to{a}}}{\left({2}-{\frac{{{x}}}{{{a}}}}\right)}^{{{\left({\tan{{\frac{{\pi{x}}}{{{2}{a}}}}}}\right)}}}\) I

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