Problem 52 Hard Difficulty

Consider the function $ f(x) = \tan \frac{1}{x} $.

(a) Show that $ f(x) = 0 $ for $ x = \frac{1}{\pi}, \frac{1}{2\pi}, \frac{1}{3\pi}, ... $

(b) Show that $ f(x) = 1 $ for $ x = \frac{4}{\pi}, \frac{4}{5\pi}, \frac{4}{9\pi}, ... $

(c) What can you conclude about $ \displaystyle \lim_{x \to 0^+}\tan \frac{1}{x} $?

Answer

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Video Transcript

This is problem number fifty two of the Stuart Calculus eighth edition Section two point two. Consider the function F of X equals tension of one of REX Party show that f of X is equal to zero for X equals one over Pi X equals one over two Pi X equals one over three play and so on. We're going to take this directly, take the first number of the second or the third number, and plug it into the function to see what we get for one over pine. We'LL have tension of one divided by one over pi, and this is actually the same as tension are pie and pull him into a calculator. Tenant of Pi is equal to zero if we do this for one over to Pine. So let's say we had a to here. Notice that that reciprocal or the reciprocal over this reciprocal gives us two here, and if we plug that into a calculator again, we should also get zero. So the reason that we get zero for pie for two pie and then eventually for three pie when we played into the ten in function. That tension is a periodic function and it has a period of pie. Therefore, every pipe, every multiple a pie results and attention equal to zero. And we are going to answer or we're going to confirm party. Ah, for many different values of X. We're going to see that the function if X is equal to zero for any X that looks like that has the form one over and pie and being a number on insecure, positive integer such as one event is one one over pi. That's the first number and is to one or two pi and so on. So this encompasses all the X values that make our function of X equals zero for Part B. We're going to show in a very similar fashion. Why, for all these ex values f of X equals one. Okay, we're going to take disfunction, tangent and of one over X, and we're going to play in the first number, which is four over pine. Since this is ah, reciprocal or affection, we end up getting tendon of the reciprocal lava forever pie or power form and into a calculator. If we plug intention of the pirate for we should get the valiant one. If we continue this better for five point is for over five pie the reciprocal bean five pi over four And this value any calculator still gives you the value of one again because the difference between power for and pipe arrow for is a pie and pie is the period of attention. And so every pie and added on to the input of the tangent function will still give you the same answer. And in this case, it is one. So we're going to formally right to that. Affects equals one for many values of X that have this form for over pie which is a first term If n is equal to one yeah, for over five point and and equals two for overnight, by and so forth. So we have answered parts A and parts be and then we're going to use those solutions too. Confirmed part. See, So what can include about the limit as experts zero from the right of detention? I want a wreck. So what we want to see here is how to we have ex parte zero because that's what the limit is about. Ex approaching zero. And since we have two cases one from party one from RP. We want to consider cream the type of numbers that approach zero for X because there are two types of excess. We see that if n if n approaches infinity here, the denominator will be a very large number. Infinity times high, Wonderful to write that is zero to that works, if any parts infinity over here. This denominator also is very large for David Obey. An infinite number is also zero. So we've confirmed that is an approaches Infinity experts zero And here's where the issues stands. Even though both of these x values approach zero, we see that these two x values are conditional our conditions for two separate F values. So this set of X values produce f of X equals zero. This set of X values produced out of X equals to one when and precious affinity experts zero. But it is uncertain what f is equal to right after and not be equal two zero and one. Let me write it at zero or one at the same time. That leads to a contradiction. And what we have at X equals here was actually at singularity and So we conclude because of these conflicting values of F as expression zero that this limit is not I exist.