Determine the following limits.

Question

\lim_{θ \to 0} \frac{2 + \sin θ}{1 - \cos^{2} θ}

Accepted Answer

Welcome back, everyone. Find the limit. Limit as x approaches divided by 2 of cosine of x minus 1 divided by 1 minus squad of X, and we're given for answer choices. A says infinity, B0, C negative infinity, and D does not exist. So let's rewrite the limit. We're given a limit as x approaches. Pi divided by 2, our function is cosine of x minus 1. Divided by 1 minus squared of x. We begin our problem by direct substitution which gives us cosine. Of pi divided by 2 minus 1. And in the denominator, we have 1 minus sin squared of pi divided by 2. Now, cosine at pi divided by 2. Gives us a value of 0, so we get 0 minus 1, which is -1. And in the denominator we get 1 minus squad of pi divided by 2. Sin of pi divided by 2 is 1, so we're subtracting 1 minus 1, right? And essentially we have 0. Whenever we have a number divided by 0, this gives us some sort of infinity, but to make sure that we understand whether we are approaching 0 from the left or from the right. We have to factor out the expression, so let's go ahead and do that. So we get limit as x approaches divided by 2. In the numerator we have cosine of x minus 1, and in the denominator, well we have to recall that sine square x plus cosine square x is equal to 1 based on the Pythagorean identity. So 1 minus squared x. Is going to be equal to cosine squad of x. We're basically subtracting signs square the bags from both sides. So now our denominator becomes cosine squared of X based on the Pythagorean identity. And what we're going to do is once again perform direct substitution. We already know that in the numerator we get -1. For the denominator, it is very easy to tell that we are approaching 0 from the right, and this is because we have the square of cosine. So even if we're approaching 0 from the left, the square is going to become positive. Whenever we divide a negative number by a positive number, the answer is going to correspond to a negative number. And in this case, this is negative infinity because we have a number divided by 0, so the limit is infinity. And we can conclude that the correct answer to this problem corresponds to the answer to C negative infinity. Thank you for watching.

Determine the following limits.

$\lim_{θ \to 0} \frac{2 + \sin θ}{1 - \cos^{2} θ}$

Key Concepts

Limits

Trigonometric Functions

Indeterminate Forms

Watch next