Squeeze theorem trig question. Please and thanks

[nukeman]

Homework Statement

Ok, you can see the question and how far I got from the image.

Lim x--> infinity

So, let's take the LEFT side: how does 3x-1 / x turn into just 3?

I know the answer is 3, but from where you can see in the pic, that is where I get stuck, and don't know how to go on from there.

THanks!

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

hi nukeman!

So, let's take the LEFT side: how does 3x-1 / x turn into just 3?

the 1 becomes insignificant compared to 3x as x -> ∞, so that leaves you with 3x/x, which is 3

alternatively: (3x-1)/x = 3 - 1/x, and 1/x -> 0 as x -> ∞, so 3 - 1/x -> 3

 

[nukeman]

How does the 1 become insignificant? That is what I am having trouble understanding. :(

Thanks for the reply :)

 

[tiny-tim]

How does the 1 become insignificant?

because the 3x becomes 30000 then 30000000000 then 300000000000000 and so on, and the tiddly little 1 doesn't matter much any more

 

[SammyS]

Homework Statement

Ok, you can see the question and how far I got from the image.

Lim x--> infinity

So, let's take the LEFT side: how does 3x-1 / x turn into just 3?

I know the answer is 3, but from where you can see in the pic, that is where I get stuck, and don't know how to go on from there.

THanks!

It generally helps us, if you state the problem you're trying to solve.

In this case it appears to be:

Find $\displaystyle \lim_{x\,\to\,\infty} \frac{3x+\cos^2(100x)}{x}\ .$​

By the way: $0\le \cos^2(100x)\le 1\ .$

 

[nukeman]

It generally helps us, if you state the problem you're trying to solve.

In this case it appears to be:

Find $\displaystyle \lim_{x\,\to\,\infty} \frac{3x+\cos^2(100x)}{x}\ .$​

By the way: $0\le \cos^2(100x)\le 1\ .$

why is it 0≤ instead of -1?

 

[SammyS]

why is it 0≤ instead of -1?

I meant to ask you !

 

[nukeman]

limit as x -> infinity of [ 3x + cos^{2}(100x)] / x

I thought it would -1 ≤ cos^2(100x) ≤ 1

since a cos has amplitude of -1 and 1 ?

 

[SammyS]

limit as x -> infinity of [ 3x + cos^{2}(100x)] / x

I thought it would -1 ≤ cos^2(100x) ≤ 1

since a cos has amplitude of -1 and 1 ?

Right regarding cos(x).

How about cos²(x)? Is that ever negative?

 

[nukeman]

wow, how did I NOT know that! :(

Ok thanks... I got that now.

So the left side would be 3x/x correct? so, then I would put each term over x, leaving 3 and 1 (3 * 1 = 3) correct?

 

[SammyS]

wow, how did I NOT know that! :(

Ok thanks... I got that now.

So the left side would be 3x/x correct? so, then I would put each term over x, leaving 3 and 1 (3 * 1 = 3) correct?

Please write the expression that you're trying to describe .