Saturday, January 19, 2019

calculus - Determine limxto0fracxsinxx3=frac16, without L'Hospital or Taylor



How can I prove that limx0xsinxx3=16



without using L'Hospital or Taylor series?




thanks :)


Answer



Let L=limx0xsin(x)x3. We then have
L=limy03ysin(3y)27y3=limy03y3sin(y)+4sin3(y)27y3sin(3y)=3sin(y)4sin3(y)=limy03y3sin(y)27y3+427limy0sin3(y)y3=327L+427
This gives us 24L = 4 \implies L = \dfrac16


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