Evaluate the following limit. lim_{xrightarrow0}frac{sin ax-tan^{-1}ax}{bx^3}

facas9 2021-01-13 Answered
Evaluate the following limit.
limx0sinaxtan1axbx3
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Expert Answer

Adnaan Franks
Answered 2021-01-14 Author has 92 answers
The given expression is
limx0sinaxtan1axbx3
Using Taylor series the expansion of sin x and tan1x
sinx=k=0(1)kx2k+1(2k+1)!
tan1x=k=0(1)kx2k+12k+1
limx0sinaxtan1axbx3
=limx0k=0(1)k(ax)2k+1(2k+1)!k=0(1)k(ax)2k+12k+1bx3
=limx0k=0(1)k(ax)2k+1(1(2k+1)!12k+1)bx3
=limx0[(1)0(a)20+1(x)202(1(20+1)!120+1)b+(1)1(a)21+1(x)212(1(21+1)!121+1b+(1)2(a)22+1(x)222(1(22+1)!122+1b+...]
=limx0[ax2(11!11)ba3x0(13!13)b+a5x2(15!15)b+...]
limx0[0a3(16)b+a5x2(15!15)b+...]
limx0[0+a36b+0]
=a36b
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Jeffrey Jordon
Answered 2022-04-01 Author has 2087 answers

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