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Problem 38 Medium Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{\sin^2 x}{x^2 + 1} $

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Video Transcript

this problem Number thirty eight over this tour Calculus eighth edition, Section two point six. Find the limit or show that it does not exist. The limit is experts in committee of Saints critics over on any X squared plus one and the way we're going to purchase them, it is actually way using a squeezed here. Um, and the squeezed serum is, ah, one that we states Let's say that we have a function that is in between two other functions that's called a G and H that is this function is between the two functions on the limits. Ah, g First, you're in condition next pretty name. If this function limited, the function G is equal to the limit as expert as a of age, then overall, this will also be the limit for aftereffects. Therefore, that is approach we're going to take. We're gonna begin with Ah, the fact that the sine function same score function. Ah, it's value always lies between zero and one. Dividing by X squared over one still gives a zero on the left hand side in the middle will have since predicts over X squared, plus one on the right side. We have won over Expert was winning. Now what we do is we take the limits book science or of each term the limit as experts is infinity. Now, every to this, we can't zero for less side. On the right side, we have the limit as explosions. Affinity of this function here one or expert was one. In this function, the denominator increases very largely towards infinity. And then the entire affection definitely goes towards zero as one over a very large number A approaches zero. So since the limit on the left equals air limit on the radical zero such as we have it here, then this limit on the inside must also equal zero. Therefore, we have prevented that the limit is expressions infinity for dysfunction. The sense critics Dr X Squared plus one s P zero through the squeeze Dear more the sandwich there