Evaluate lim x→0 x + 1/ 1 −cos x.

Question

Accepted Answer

Welcome back, everyone. In this problem, we want to solve for the limit of X2 + 1 divided by 1 minus +2 X as X approaches 0. A says it's infinity, B negative infinity, C 0, and D says it's 1. Now how can we figure out the limit of this function as x approaches 0? Well, one thing can help is if we recall the identity that tells us that 1 minus + 2 X is equal to 2 sin x squared. So in that case, OK, then we could rewrite our function as the limit. As x approaches 0 of X2 + 1 divided by 2 square x. Now we're going to solve the limit, we also know that the limit exists if the left hand limit equals the right hand limit, and both of them are the same. In other words, OK, if, let me just put this as a box here, OK, if. The limit as x approaches 0 from the left of our function. is equal to the limit as x approaches 0 from the right of our function, which is equal to some value l, then that means our limit as X approaches 0 for the function is going to be equal to that value L and it also exists. So we need to find a right and left hand limit for our function to help us solve. So let's first find the limit from the right, OK. No, from the right. From the right, then the limit, OK, as X approaches 0 from the right of X2 + 1 divided by 2 sin square x, OK. is going to be equal to 02, we can substitute 0 in our numerator, so that's 02 + 1. And now in our denominator based on what we know about the sine function, as X approaches 0, sine X is also going to be approaching 0 or a small value, OK? A very small value. So we could write this as it approaching 0 from the right. No, 02 + 1 equals 1 and 1 divided by a very small value is going to be equal to infinity because the smaller the value gets, the larger the result or the larger the quotient will be. Therefore, this limit as x approaches zero from the right equals infinity. Now, from the left, OK. Then I limit And X approaches 0 from the left. It's quite, it, it's basically going to work out to the same thing as if it were approaching it from the right because no numerator is going to be one and the sign function is also going to be approaching a very small value from the left, OK? And no, once we divide one by the smaller the value gets, then this is going to be equal to infinity. So now since the left hand limit. is equal to the right hand limit which is infinity. That means the limit as x approaches 0 for our function X2 + 1 divided by 1 minus + 2 X is going to be equal to infinity. Therefore, if we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Evaluate lim x→0 x + 1/ 1 −cos x.

Key Concepts

Limits

Indeterminate Forms

Trigonometric Limits

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