lim x tends to 0( tanx-sinx)/x^3

evaluate the limit function
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lim at x gotu 0 [tan(x) -sine(x)] / x^3... tan(x)=x+x^3/3... sine(x)=x-x^3/6... tan - sine=x^3 / 6... (tan -sine)/x^3=1/6
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The series expansions are,

tan(x) = x + x^3/3 + 2x^5/15 + 17x^7/315 + ...

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

tan(x) - sin(x) = x^3(1/3 + 1/3!) + x^5(2/15 + 1/5!) + x^7(17/315 + 1/7!) + ...

f(x) = [tan(x) - sin(x)]/x^3 = 1/2 + 17x^2/120 + 13x^4/240 + ...

Lim of f(x) as x tends to zero is: 1/2

 

by Level 11 User (81.5k points)

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