Evaluate each limit.

Question

\lim_{x \to π} \frac{\cos^{2} (x) + 3 \cos (x) + 2}{\cos^{} (x) + 1}

Accepted Answer

Welcome back, everyone. Determine the limit. Limit as X approaches to pi of two cosine square x minus 5 cosine of X plus 3 divided by cosine of X minus 1, and we're given 4 answer choices A says 0, B1, C -1, and D-2. So let's rewrite the limit. It says limit as X approaches to pi. In the numerator, we have to cosine squared of X minus 5 cosine of X plus 3. In the denominator we have cosine of x minus 1. And we begin with direct substitution. This gives us 2. Cosine squared of 2 pi. -5 cosine. Off to pine. Plus 3 in the denominator, we end up with cosine of 2 pi. -1 And now we can notice that this is an indeterminate form, we end up with 0 divided by 0. Now, what we want to do is potentially eliminate a possible hole, right? So we can notice that the numerator. Corresponds to a quadratic. equation if we We write it as follows. We can say that Z is equal to cosine of X, and then we end up with 2 z squared minus 5 Z. Plus 3. And now we can see that this is a parabola right in terms of Z and we can factor it out because we have a quadratic quinomial. And factoring it out, we end up with 2. See -3 multiplied by Z minus 1. And since he is cosine, well, we can substitute cosine back, right? But now considering our limit, if we go back to the limit itself, let's rewrite the numerator in its factored form. We get 2 cosine of x minus 3 multiplied by cosine of x minus 1 divided by cosine of X minus 1, and we indeed have a whole. Because there is a common factor in the numerator and the denominator of the rational function. And now we can apply the direct substitution for the remainder of the function, so we get to cosine. Of Ty. Minus 3. Now cosine of 2 pi is equal to 1, so we simply have 2 minus 3, which is -1, and that's our final answer which corresponds to the answer choice C. Thank you for watching.

Evaluate each limit.

$\lim_{x \to π} \frac{\cos^{2} (x) + 3 \cos (x) + 2}{\cos^{} (x) + 1}$

Key Concepts

Limits

Trigonometric Functions

Indeterminate Forms

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