Question

Help with limits please

Answer 1

Whats the question?

Answer 2

Sorry was logging in again.
\[\lim_{x \rightarrow -infinity}\cos\frac{ 1 }{ x }\]

Answer 3

No problem!

Alright so we are to see what this limit comes out to as x approaches -infinity

If I take a large negative number....like -100,000 and plug that in for 'x' what will I get?

Well \[\large \frac{1}{-100000} = -0.00001\]

OR if I go 1 step farther and make that -1,000,000 that will give us \(\large \frac{1}{-1000000}=-0.000001\)

Meaning we are getting CLOSER and CLOSER to zero each time we go closer and closer to -infinity

So basically....what is \(\large cos(0)=?\)

Answer 4

- infinity right?

Answer 5

or zero?

Answer 6

Neither, plug it into a calculator :)

Answer 7

one :o

Answer 8

There we go! As your 'x' gets closer and closer to -infinity...your limit will evaluate to 1!

Answer 9

oh ok thanks so much :D

Answer 10

is there a way to understand like why cos equals 0 or like trig identities,
i think they are called, in general?

Answer 11

Definitely *For future reference* look up the unit circle! It will help with future trig applications!

Answer 12

\(\lim\limits_{x \rightarrow -\infty}\cos\frac{ 1 }{ x }\)

\(=\cos (\lim\limits_{x \rightarrow -\infty} \frac{ 1 }{ x })\)

as cos is continuous

Answer 13

so i would just memorize it then?

Answer 14

Well it's more of something to work through...memorization sure will get you through some things but actually understanding it will make it so you dont have to think "Crap was it sin(x)=0 at 90 degrees or at 180 degrees or was that cosine??

Knowing how to implement it will make it SO much easier going forward!

Answer 15

I guess my question is, how to actually understand it. I try to read my text book, but I eventually just end up getting lost most of the time.

Answer 16

Well like what specifically? This? Like knowing when cos(x) = 0? Or sin(x) = 0? Stuff like that?

Can always walk you through examples of any problems you have with questions!

Answer 17

Yea I guess its most of the stuff I didn't end up understanding in precal tbh. I always get stumped if I see like "what is sin2x/x as x approaches infinity"

Answer 18

which is actually my next question i have to do coincidentally

Answer 19

Alright that's fine

So for now...focus on JUST the limits...nothing about trig

Here we have \(\large sin(\frac{2x}{x}) \)

Don't plug anything in yet...just think, can ANYTHING about this be simplified at all? What is your first instinct?

Answer 20

cross out the x's

Answer 21

Great! So now we're just down to \(\large sin(2)\)

So starting from the original question...now we just have \(\large \lim_{x \rightarrow \infty} sin(2) \)

Answer 22

Now guess what...

The question: What happens as x goes to infinity?
The answer: What x? There is none!

So literally all we have from: \(\large \lim_{x\rightarrow \infty} sin(\frac{2x}{x})\) is just \(\large sin(2)\)

Answer 23

so there's no limit because the x is factored out?

Answer 24

Oh theres a limit...but it doesnt depend on 'x' at all!

Answer 25

oh ok so its sin(2)? Am i suppose to know what that is?

Answer 26

Oh hell no XD That is something you plug into your calculator! :)

Answer 27

Oh ok. Are there like "easy" ones that are helpful to know? I remember in precal there was cos(0)=1 and i think there might be others, but i havent learned them

Answer 28

Well since you took precalc, you know what the basic Cos(x) and Sin(x) graphs look like

Answer 29

Cos:

Answer 30

Sin:

Answer 31

From those graphs you can see at which points each one would equal 0

Answer 32

Also, you will (Or should have already learned about) special triangles

30-60-90 triangles and 45-45-90 triangles

Those will give you your answers for things like cos(30) cos(45) sin(60) etc...

Answer 33

oh ok thanks so much for the info! I do remember going over them, but again, I haven't fully learned them... This is it for now as my school day is about to end in a min. But thanks again for the explanations and info :D

Answer 34

No problem! And we're always here to help if you have any other questions!