Use Taylor's theorem to evaluate the following limits. lim_{xrightarrow0}frac{3sin^2(x)+2sin^4(x)}{3xtan(x)}

Alyce Wilkinson 2021-02-23 Answered
Use Taylors
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Expert Answer

svartmaleJ
Answered 2021-02-24 Author has 92 answers
Evaluate limit using Taylor’s theorem.
Given:
limx03sin2(x)+2sin4(x)3xtan(x)
Taylor series expansion of trigonometric functions,
sin2x=x2x43+2x645x8315+...
sin4x=x42x63+x8534x10945+...
tanx=x+x33+2x515+17x7315+62x92835+...
Substitute the series,
limx03(x2x43+2x645x8315+...)+2(x42x63+x8534x10945+...)3x(x+x33+2x515+17x7315+62x92835+...)
limx0(3x23x43+3(2x6)453x8315+...)+(2x42(2x6)3+2x852(34x10)945+...)(3x2+3x43+32x615+317x8315+362x102835+...))
Divide x2 in both numerator and denominator,
limx>0(33x23+3(2x4)453(x6)315+...)+(2x22(2x4)3+2x652(34x8)945+...)(3+3x23+(32x4)15+(317x6)315+(362x8)2835+...)
Apply limit, x tends to 0
=3/3
1
Result: 1
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