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Use l’Hôpital’s rule to find lim(csc(x) - cot(x)) as x -> 0.

This question relies partially on remembering the trigonometric identities, csc(x) = 1/(sin(x)) and cot(x) = 1/(tan(x)) = cos(x)/sin(x). Plugging these values into the equation, we want to find Lim((1-cos(x))/sin(x)) as x -> 0.

l'Hôpital's rule states that we can differentiate the top and bottom of the fraction separately and the limit will stay the same (note: we are not differentiating the fraction as a whole, this requires the quotient rule, but instead differentiating the numerator and denominator individually).

We do so to get that the limit is now Lim((sin(x)/cos(x)) as x -> 0 which is just 0. Therefore, the limit of the original equation is 0.

12 months ago

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