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Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.$\lim _{x \rightarrow 0} \cot 2 x \sin 6 x$
3
Calculus 1 / AB
Chapter 4
Applications of Derivatives
Section 3
L'Hospital's Rule: Comparing Rates of Growth
Differentiation
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Okay, so here we have the limit. As act approaches zero of co tangent of two x times Sign of six X. So we will start by rewriting this into a fraction form. So we have the limit as X approaches zero and co tangent is, of course, just one over tangent. So this is really Ah sign of six X over Tangent of two ex. Okay. And when we use this form and we evaluate the the limit, we would get sign of zero over a tangent of zero. And that's going to be 0/0 so we can fly Low petals rule. So then this is gonna be equivalent to the limit. As X approaches, zero will take the derivative of sign of six X with respect to X, and that is gonna be six times co sign of six X and then the derivative of tangent of two X is going to be two times seeks squared of two X, and perhaps it helps toe put those in the brackets for clarity. Ah, and so now we can evaluate this one as well. When we evaluate it, we will get six times co sign of zero over, too. Time. Six squared of zero. Okay, this is gonna give us six times 1/2 times one, which is equal to three. That's the solution. Really. The trick is just in rewriting it, so it's in a form where little's rule
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