Problem 67 Hard Difficulty

Find $ \displaystyle \lim_{x \to \infty} f(x) $ if, for all $ x > 1 $, $$ \frac{10e^x - 21}{2e^x} < f(x) < \frac{5\sqrt{x}}{\sqrt{x - 1}} $$

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this is problem number sixty seven of the sewer calculus eighth edition Section two Going six Find the limit, its expertise, infinity of the function F if for all extra grated and won after taxes. Greater than ten either the explains twenty one divided by two edicts and less than five times discredit of ex over the square root of the quantity X minus one. So we're going to go ahead and approach this problem as a squeezed hearing problem. Recall what that means if we have a function if that is less than our function G and greater than a function each and we were able to prove that ah, the limit and sex purchase value, eh for H is equal to the limit is expert is a for Jean. Therefore, they have the same limit we'LL call Ellen. Then that would mean that the limit is extra protein for the function. F is also equal to our So this will be our approach. We already have a pretty straightforward set up. We just need to prove that the limit of the left function here is limited right function and then the answer should be consistent. If not, then we'Ll have to think of a different approach but for now we're pretty confident. So we proceed. Limited's X approaches Infinity of the first function ten either the ex minus twenty one over to E. T. X We can tobi by either the ex each term. That should simplify problem a little bit more ten minus twenty one operative X all over too. At this point, we identified through limit property limits that we can take the limit of each term individually and this term here, twenty wanna read of X approaches. Zero is X approaches infinity. So what we're left or with is the limited express infinity of ten over too ten or two, of course, gives us five and that is the answer for a limit for this left function inch of X. Now let's see what the limit is as expert is infinity of the right function G of x, five times square of X divided by the quantity sort of experts one Ah, let's see, We're going attempts to divide this crate of X right to each term Similarly, this problem a little bit the numerator now five the denominator If we divide the square root of explaining the quantity expense wanted. Get it right. Squirt of X should give us about squirt of the party one minus one over X and his expertise. Infinity, This term becomes like negligible. It approaches zero. And so our limit is equal to family this quarter to one or five. So we were able to find at the limit on this left side. As expert is, infinity is five. The limit on the right side is also equal to find that Tyrrell So we can conclude that the limit as X approaches infinity or affects, even though we're not quite sure what the function is exactly F We know that the limited Express infinity must also be five provided that it is greater than this function on less than dysfunction by the squeezed here.