Determine the following limits.
lim θ→π/2 sin^2 θ − 5 s...

Question

Determine the following limits.

lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

Accepted Answer

Welcome back, everyone. Evaluate the limit as theta approaches 0 of the function f of theta equals cosine quad minus 5 cosine theta plus 4 divided by cosine square theta minus 1. So let's write down the limit. We want to evaluate limit as the approaches. 0 Of cosine squared of theta. -5, cosine of theta plus 4. And that's divided by. Cosine squared of theta minus 1. First of all, let's apply the direct substitution. We get cosine squared of 0 minus 5 cosine of 0. Plus 4 Divided by cosine squared of 0 minus 1. And we end up with an indeterminate form which is 0 divided by 0. That said, we might have a potential hole in this rational function, and we want to factor it out. So looking at the numerator, let's suppose that Z is equal to cosine of theta. Then we can rewrite our numerator as Z2 minus 5 Z + 4. And we can factor out this trinomial. S Z minus 4 multiplied by Z minus 1. Where Z is cosine of theta. This helps us rewrite the limit as follows limit as theta approaches 0 of. Oh Theta minus 4 multiplied by cosine theta minus 1. And in the denominator we have a difference of squares which can be factored out as cosine of theta minus 1. Multiplied by cosine of theta plus one. So we have completely factored it out and we can see that there is a common factor that we can cancel out that's cosine of data minus 1. From here for the remainder we can apply direct substitution. Because we have canceled out the whole. And now we get cosine of 0 minus 4 divided by. Cosine of 0 + 1. So that's 1 minus 4, which is -3, divided by 1 + 1, which is 2. And this is a finite number. We have successfully evaluated the limit -3 halves. Thank you for watching.

Determine the following limits.
lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

Key Concepts

Limits

Trigonometric Functions

Factoring and Simplifying Expressions

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