17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0 csc 6x sin 7x

Accepted Answer

Welcome back, everyone. Apply Lato's rule to find the limit of the following function as X approaches by divided by 2. We're given our function F of X and for answer choices A says 1, B-1, C 1/3, and D 1/3. So let's write down the limit. Limit is x approaches pi divided by 2. Our function F of X is 2 of 3x multiplied by cosine of X. If we apply the right substitution, we're going to get c of 3 pi divided by 2, which is 1 divided by cosine of 3 pi divided by 2. And this is multiplied by cosine of pi divided by 2. Now we want to evaluate each term, so cosine of pi divided by 2 is equal to 0, and now cosine of 3 pi divided by 2. is equal to 0 as well, so 1 divided by that approach is infinity, right? So we get a form infinity multiplied by 0, and this form is not appropriate for Lapel's rule. So what we're going to do instead is simply rewrite this as a fraction limit. As x approaches pi divided by 2 of cosine of x divided by cosine. Of 3 acts and in this case while we get an indeterminate form 0 divided by 0 which works for lapto's rule, right. And Lapido's rule says that we can differentiate. The numerator and the denominator of the rational functions. So let's find the derivative of cosine of X, which is negative sine of X. And now we want to find the derivative of the denominator cosine of 3 X. So that would be negative sign of 3 X. We need to apply the chain rule because we have an inner function. So we're multiplying by the derivative of 3X, which is simply 3, right? And now from here we want to once again apply the right substitution. Notice that we can cancel out the negative signs. We can also write 3 in front. And the denominator, and this gives us sign. Of I divided by 2, divided by. We Sign of 3 pi divided by 2. So now let's go ahead and evaluate each term. So for the numerator, we get 1, that's sin of pi divided by 2. Now sign of 3 pi divided by 2 is -1 and -1 multiplied by 3. I-3. So overall we now get the finite number. It is -1/3 and therefore the correct answer choice this problem corresponds to the answer choice D. Thank you for watching.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0 csc 6x sin 7x

Key Concepts

Limits

L'Hôpital's Rule

Cosecant and Sine Functions

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