Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

Accepted Answer

Welcome back, everyone. Evaluate the limit using Lapital's rule. Limit as X approaches 0 of LN of C X divided by X2. Let's begin this limit by Direct substitution. So we're going to get limit as X approaches 0 of LN of C X divided by X squared. Let's remember that sequent is 1 divided by cosine. So if we use the direct substitution, we're going to take 1 divided by cosine of 0, which is 1, and LN of 1 is 0. And then we're dividing by 0 squared, so we get an indeterminate form 0 divided by 0, meaning oppato's rule can be applied immediately. According to Lapital's rule, we want to take the derivative of the numerator, so that would be the derivative of LN of C and X, divided by the derivative of the denominator. Let's focus on each derivative for the numerator. We have a composite function, so we want to apply the chain rule. Remember that first of all, we're differentiating the natural log with respect to the inner function to get 1 divided by sequent X, and we're multiplying by the derivative of sequent X according to the chain rule. The derivative of 2 is 2 X tangent X. And I noticed that we can cancel out sequent, right? So we basically get tangent X. In the numerator and for the denominator we get 2 X using the power rule. Once again, we attempt the direct substitution, tangent of 0 is 02 multiplied by 0 is 0. So we still have the same in the terminate form, which means that we're going to apply Lapito's rule once again. We want to differentiate tangent X, right? So we take the derivative of tangent X, which is squared X. And we want to take the derivative of 2 X. We can factor 2 and differentiate X, the derivative of X is 1, so we simply get 2. Once again, we attempt the right substitution. We can factor out one half and we basically want to evaluates and squad of 0, that's 1 divided by cosine squared of 0. Cosine of 0 is 1, so we get 1 divided by 1 squared which is 1, multiplied by 1/2. This gives us a limit of 12, which is our final answer. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²