Problem

Use a graph to estimate the equations of all the …

08:50

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

Check back soon!

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Discussion

You must be signed in to discuss.

Top Calculus 1 / AB Educators

Watch More Solved Questions in Chapter 2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

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.

Daniel J.

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Problem

Use a graph to estimate the equations of all the …

Problem 52 Hard Difficulty

Answer

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Catherine R.

Kayleah T.

Joseph L.

Calculus 1 / AB Bootcamp

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 2

Video Transcript

Calculus: Early Transcendentals

Grace H.

Catherine R.

Kayleah T.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview