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.

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

Accepted Answer

Welcome back, everyone. Evaluate by Lapital's rule limit as theta approaches pi divided by 4 of tangent of theta minus 1 divided by theta minus pi divided by 4. A says 2, B1, C square root of 2, and D 1/2. For this problem, let's rewrite the limit. We have limit as theta approaches pi divided by 4 of tangent of theta minus 1 divided by theta minus pi divided by 4. We begin by substituting theta equals by divided by 4 for every theta to get tangent of. Pi divided by 4 minus 1, divided by pi divided by 4 minus pi divided by 4. Tangent of pi divided by 4 is 1, so we get 0 in the numerator because that's 1 minus 1. In the denominator, i minus 4 minus. I'm sorry, i divided by 4 minus pi divided by 4 is also 0. So we get an indeterminate form 0 divided by 0, which is suitable for Lapto's rule. Now, we can apply Lapito's rule, which says that we can differentiate the numerator and the denominator separately, right? So, we're going to take the derivative of tangent of the minus 1. And divided by the derivative of theta minus divided by 4. Now, let's take each derivative. We're going to get limit as theta approaches by divided by 4. The derivative of tangent is squad, so we're going to get s and squad of data. The derivative of -1 is 0 because it's a constant. In the denominator, the derivative of theta is 1. And the derivative of negative pi divided by 4 is 0. It's a constant. So now we have limit as data approaches by divided by 4 of second squad of data. And again, we're going to perform direct substitution. Sent is 1 divided by cosine, so we're getting 1 divided by cosine squared off by divided by 4, right? Now let's remember. The value of cosine of pi divided by 4. Now, we can drop the limit, right? Because we are performing the direct substitution. So that would be one divided by. The value of cosine of pi divided by 4 is square root of 2 divided by 2, and we are squaring it, right? So now this is going to be a finite value. We can draw brackets and we can basically rewrite one divided by. 2 divided by 4, right, or basically 1 divided by 12. Which is equal to 2. So the final answer for this problem is option A. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

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

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)