Evaluate limit using Taylor’s theorem. Given: limx→03sin2⁡(x)+2sin4⁡(x)3xtan⁡(x) Taylor series expansion of trigonometric functions, sin2⁡x=x2−x43+2x645−x8315+... sin4⁡x=x4−2x63+x85−34x10945+... tan⁡x=x+x33+2x515+17x7315+62x92835+... Substitute the series, limx→03(x2−x43+2x645−x8315+...)+2(x4−2x63+x85−34x10945+...)3x(x+x33+2x515+17x7315+62x92835+...) limx→0(3x2−3x43+3(2x6)45−3x8315+...)+(2x4−2(2x6)3+2x85−2(34x10)945+...)(3x2+3x43+3⋅2x615+3⋅17x8315+3⋅62x102835+...)) Divide x2 in both numerator and denominator, limx−&gt;0(3−3x23+3(2x4)45−3(x6)315+...)+(2x2−2(2x4)3+2x65−2(34x8)945+...)(3+3x23+(3⋅2x4)15+(3⋅17x6)315+(3⋅62x8)2835+...) Apply limit, x tends to 0 =3/3 ⇒1 Result: 1

Key to "Use Taylors"

Alyce Wilkinson

Answered question

2021-02-23

Use Taylors

Answer & Explanation

svartmaleJ

Skilled2021-02-24Added 92 answers

Evaluate limit using Taylor’s theorem.
Given:

lim_{x \to 0} \frac{3 \sin^{2} (x) + 2 \sin^{4} (x)}{3 x \tan (x)}

Taylor series expansion of trigonometric functions,

\sin^{2} x = x^{2} - \frac{x^{4}}{3} + \frac{2 x^{6}}{45} - \frac{x^{8}}{315} + . . .

\sin^{4} x = x^{4} - \frac{2 x^{6}}{3} + \frac{x^{8}}{5} - \frac{34 x^{10}}{945} + . . .

\tan x = x + \frac{x^{3}}{3} + \frac{2 x^{5}}{15} + \frac{17 x^{7}}{315} + \frac{62 x^{9}}{2835} + . . .

Substitute the series,

lim_{x \to 0} \frac{3 (x^{2} - \frac{x^{4}}{3} + \frac{2 x^{6}}{45} - \frac{x^{8}}{315} + . . .) + 2 (x^{4} - \frac{2 x^{6}}{3} + \frac{x^{8}}{5} - \frac{34 x^{10}}{945} + . . .)}{3 x (x + \frac{x^{3}}{3} + \frac{2 x^{5}}{15} + \frac{17 x^{7}}{315} + \frac{62 x^{9}}{2835} + . . .)}

lim_{x \to 0} \frac{(3 x^{2} - \frac{3 x^{4}}{3} + \frac{3 (2 x^{6})}{45} - \frac{3 x^{8}}{315} + . . .) + (2 x^{4} - \frac{2 (2 x^{6})}{3} + \frac{2 x^{8}}{5} - \frac{2 (34 x^{10})}{945} + . . .)}{(3 x^{2} + \frac{3 x^{4}}{3} + \frac{3 \cdot 2 x^{6}}{15} + \frac{3 \cdot 17 x^{8}}{315} + \frac{3 \cdot 62 x^{10}}{2835} + . . .))}

Divide

x^{2}

in both numerator and denominator,

lim_{x - > 0} \frac{(3 - \frac{3 x^{2}}{3} + \frac{3 (2 x^{4})}{45} - \frac{3 (x^{6})}{315} + . . .) + (2 x^{2} - \frac{2 (2 x^{4})}{3} + \frac{2 x^{6}}{5} - \frac{2 (34 x^{8})}{945} + . . .)}{(3 + \frac{3 x^{2}}{3} + \frac{(3 \cdot 2 x^{4})}{15} + \frac{(3 \cdot 17 x^{6})}{315} + \frac{(3 \cdot 62 x^{8})}{2835} + . . .)}

Apply limit, x tends to 0

= 3 / 3

\Rightarrow 1

Result: 1

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