Use Taylor's theorem to evaluate the following limits. lim_{x->0}({xsin(x)-x^2)/(cos(x)-1+(x^2)/(2))

Daniaal Sanchez 2020-10-26 Answered

Use Taylor's theorem to evaluate the following limits. limx0xsin(x)x2cos(x)1+x22

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Expert Answer

Demi-Leigh Barrera
Answered 2020-10-27 Author has 97 answers

Evaluate limit using Taylor’s theorem.
Given:
limx0xsin(x)x2cos(x)1+x22
Taylor series expansion of trigonometric functions,
sinx=xx33!+x55!x77!+...,
Substitute the series,
limx0xxx33!+x55!x77!+...1x22!+x44!x66!+...1+x22
limx0(x2x43!+x65!x87!+...)x21x22!+x44!x66!+...)1+x22)
Cancelling the term,
limx0x43!+x65!x87!+...x44!x66!+...
Divide x4 in both numerator and denominator,
limx013!+x25!x47!+...14!x26!+..
Apply limit, x tends to 0.
13!14!
4!3!
4321321
4

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