Triple kill

Calculus Level 2

Evaluate lim ⁡ x → 0 x 3 x − sin ⁡ ( x ) \displaystyle \lim_{x \to 0} \frac{x^3}{x - \sin(x)} .


The answer is 6.

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Calvin Lin by Calvin Lin

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2 solutions

To evaluate this limit, we will use L'Hôpital's rule. Using the rule for the first time will give us (3(x^2))/(1-cosx). When x=0, the quotient will be 0/0, so we use L'Hôpital's rule again. This will give us (-6x)/sinx. When x=0, the quotient will be 0/0, so we use L'Hôpital's rule for the third time to get 6/cosx. When x=0, the quotient is 6. Therefore the limit as x approaches 0 is 6

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what the "l'hôpital's" rule says

Chaïmae Mezzat - 7 years, 2 months ago

Since sin(x) = x - (x^3)/6 when x approaches 0 . then the limit is 6 when x approaches 0

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