Question

Need help with limits!

Answer 1

\[\lim_{x \rightarrow \infty} (x+sinx)/(x+cosx)\]

Answer 2

Um an explanation would be nice l0l

Answer 3

Multiply top and bottom by 1/x
\[\frac{ x + sinx }{ x + cosx } = \frac{ \frac{ x }{ x }+\frac{ sinx }{ x } }{ \frac{ x }{ x } +\frac{ cosx }{ x }}\]

Answer 4

sinx/x is 1
isnt cosx/x undefined?

Answer 5

As you go to infinity, both sinx/x and cosx/x are 0. This can be shown with squeeze theorem if needed.

Answer 6

@Concentrationalizing Oh wait I'm dumb nevermind

Answer 7

Well, both sinx/x and cosx/x work the same way if you consider a squeeze theorem proof. Both sinx and cosx are bounded between are -1 and 1. So we can say that this inequality is true:
\[-1 \le cosx \le 1\]

Lol, oh, do you see what I mean now then?

Answer 8

Yeah I understand what you are getting at. So it seems like the answer would be 1.

Answer 9

That would be correct :)

Answer 10

Man limits are hard. My teacher never went over this kind of stuff in class l0l

Answer 11

I find limits of trig super hard to grasp..

Answer 12

You kind of just learn what works over time. The first time I came across these the thought of multiplying by 1/x would've never come to mind, lol.

Answer 13

The problem is that my teacher went over the proofs of sinx/x and cosx/x. So I just kind of memorize them, although using the squeeze theorem to prove them makes a lot of sense!

Answer 14

Thanks for your help!

Answer 15

Hell if I remember those proofs, haha. If it's any bounded function, then squeeze theorem is always worth a shot. sinx and cosx are the perfect examples of bounded functions, so they come out really well.

Answer 16

All these theorems get me confused. Squeeze theorem, intermediate value theorem... lol
Just gotta study more :p

Answer 17

Yeah, it kind of is a lot. Anything with trig you're basically trying to force a sinx/x or cosx/x situation, so that's how I know to go there. Otherqise squeeze theorem never seemed to come up again until calc 2 or analysis. But eventually you learn about l'hopital's rule and that makes a lot of other limits easy to compute, so it doesnt really get any harder xD

Answer 18

It's Grade 12 AP Calculus, D: we finished the limits unit in a week. We are also finishing derivatives and integrals in like 2months so rip me

Answer 19

Lol, nice. Gah, I wish I had the opportunity to do something like that. I dont think my high school even offered calculus. I'm jealous of those who come out of high school and have already done multivariable calculus and linear algebra and such, lol. I'd have loved to do what you're doing back when I was in 12th grade.

Answer 20

It's fun but it's such a new concept compared to advanced functions. We were graphing simple rational functions and boom understand limits and these theorems. It feels like a big jump. Although I do enjoy learning calculus :3

Answer 21

Yeah, I think its fun, too. But limits are very important, so it has to be done. It's not really a big leap to be honest, though. When you were learning about graphing rational functions and such, horizontal asymptotes are the same as taking the limit as x goes to positive and negative infinity. Now theyre just expanding the idea and testing your algebra skills xD

Answer 22

It's just the trig limits that are screwing me over. Limits of polynomial functions are super easy though. Just when there's a function what doesn't approach to anything when it goes to infinity.. well then I have a problem xD

Answer 23

Well, if you ever come across other examples, feel free to ask. I'm sure you'll get used to what to do with them. There's usually only so many things that you can do unless your professor is just out to be mean to ya, lol.

Answer 24

Thanks a ton! appreciate your help

Answer 25

Good luck ^_^