Master the Art of Finding Limits Involving Trig with Expert Homework Help

[Torshi]

Homework Statement

Finding limit: involves trig

Homework Equations

Lim X-> 0 ((tan(x))^2 / X)

The Attempt at a Solution

I've done multiple other problems, but can't do this one, my trig is weak.

 

[SammyS]

Homework Statement

Finding limit: involves trig

Homework Equations

Lim X-> 0 ((tan(x))^2 / X)

The Attempt at a Solution

I've done multiple other problems, but can't do this one, my trig is weak.

Did you try L'Hôpital's rule ?

 

[Mark44]

A well-known limit that can be used in this problem is this one:
$$ \lim_{x \to 0} \frac{sin(x)}{x} = 1$$

 

[Torshi]

Did you try L'Hôpital's rule ?

We're not there yet in lecture. I just started College level calculus. We have so far just defined what limits are and using εδ def. of limits and I'm having a hard time understanding the rules for using epsilon and delta to prove an argument and some limit problem solving involving trig etc.

 

[Torshi]

A well-known limit that can be used in this problem is this one:
$$ \lim_{x \to 0} \frac{sin(x)}{x} = 1$$

I know this and the other one is = 0. My prof said memorize it, but I don't know how to incorporate it. I feel like I'm in a rough start for calc...I'm not bad in math per se, I've taken college alg, statistics, physics 1/2, gen chem 1/2 etc all a year ago or more, haven't taken trig or precal. I took the math placement and scored enough somehow to end up in this hard class that i need.

 

[SammyS]

I know this and the other one is = 0. My prof said memorize it, but I don't know how to incorporate it. I feel like I'm in a rough start for calc...I'm not bad in math per se, I've taken college alg, statistics, physics 1/2, gen chem 1/2 etc all a year ago or more, haven't taken trig or precal. I took the math placement and scored enough somehow to end up in this hard class that i need.

Write tan(x) in terms of sin(x) & cos(x) .

 

[Torshi]

Write tan(x) in terms of sin(x) & cos(x) .

are you implying i should need to know the identities such as 1-cosx = sinx^2 for example - not that this has to do anything with the prob? Sorry if I'm off, but I never took trig

 

[Dick]

are you implying i should need to know the identities such as 1-cosx = sinx^2 for example - not that this has to do anything with the prob? Sorry if I'm off, but I never took trig

You need to know tan(x)=sin(x)/cos(x). You do need to know a LITTLE trig.

 

[Mark44]

are you implying i should need to know the identities such as 1-cosx = sinx^2 for example

It's actually cos²(x) = 1 - sin²(x).

- not that this has to do anything with the prob? Sorry if I'm off, but I never took trig

That's something you'll need to rectify, since many of the problems you'll see will require some knowledge of basic trig identities. For starters, you need to be able to write tan(x), cot(x), sec(x), and csc(x) in terms of sin(x) and/or cos(x).

 

[Torshi]

You need to know tan(x)=sin(x)/cos(x). You do need to know a LITTLE trig.

I still can't solve it. I know sin(x)/x = 1

edit: Alright, I'll brush up on my trig identities etc. But, this HW is due on Tuesday. I'll youtube how to do that

 

[Mark44]

I still can't solve it. I know sin(x)/x = 1

No, $lim_{x \to 0} \frac{sin(x)}{x} = 1$, but that's different from what you wrote.

edit: Alright, I'll brush up on my trig identities etc. But, this HW is due on Tuesday.

 

[Torshi]

So I have written down all the identities. Would (Tan(x)^2) which is the numerator first turn into Tan^2(x) which i believe is another way of writing it.

I also have down here that Tanθ = Sinθ / Cosθ.....Also tan^2θ = Sec^2θ-1 ?

Edit: I did a similar problem and got it right. lim X--> 0 F(x) = 1-cosx / x

1.) 1-cosx / x multiplied by 1+cosx / 1+cosx
2.) 1^2 - cosx^2 / x (1+cosx) then becomes sin^2x / x(1+cosx)
3.) 1 x 0 = 0

 

[Torshi]

So far I'm assuming I have:

1.) tan^2x = Sec^2x - 1

2.) Sec^2x -1 / x X Sec^2x+1 / Sec^2+1

3.) Sec^4 -1 / x (sec^2x+1)

?? good so far?

edit: I think I'm doing it wrong

 

[haruspex]

Go back to tan = sin/cos, sin(x)/x tends to 1 as x tends to zero. What do you think cos(x) tends to as x tends to 0?

 

[Torshi]

Go back to tan = sin/cos, sin(x)/x tends to 1 as x tends to zero. What do you think cos(x) tends to as x tends to 0?

I think the answer is zero I believe for my overall problem

 

[haruspex]

Yes, but why?

 

[Torshi]

Yes, but why?

I don't know I'm so confused, this crap has been bugging me all day. It's been a while since I've done all this.

All my prof said was that lim x-->0 sinx/x = 1 and lim x-->0 1-cosx/x = 0 . This is calc so he just assumes we know it, he didn't review any precalc or trig obv. He just speeds right through.

I wrote down all the trig identities such as tanx=sinx/cosx etc etc and cos^2x + sin^2x =1

I know these things are all related. In one of my posts I figured out an answer due to trig identities. I can't for this one... I'm sure it's very simple, I just can't see it.

I don't see how I can relate (tan(x)^2)/x to any of them...

 

[Dick]

I think the answer is zero I believe for my overall problem

Yes, it is. Can you explain why?

 

[Dick]

I don't know I'm so confused, this crap has been bugging me all day. It's been a while since I've done all this.

All my prof said was that lim x-->0 sinx/x = 1 and lim x-->0 1-cosx/x = 0 . This is calc so he just assumes we know it, he didn't review any precalc or trig obv. He just speeds right through.

I wrote down all the trig identities such as tanx=sinx/cosx etc etc and cos^2x + sin^2x =1

I know these things are all related. In one of my posts I figured out an answer due to trig identities. I can't for this one... I'm sure it's very simple, I just can't see it.

I don't see how I can relate (tan(x)^2)/x to any of them...

tan(x)^2/x=(sin(x)/x)*(sin(x)/cos(x)^2). What's the limit of each of those two factors?

 

[haruspex]

I don't know I'm so confused, this crap has been bugging me all day. It's been a while since I've done all this.

All my prof said was that lim x-->0 sinx/x = 1 and lim x-->0 1-cosx/x = 0 .

That had better be lim x-->0 (1-cosx)/x = 0 .

I wrote down all the trig identities such as tanx=sinx/cosx etc etc and cos^2x + sin^2x =1

Do you know what cos(0) is? Given that, and knowing that sin(x)/x -> 0, and sin(x) = cos(x)tan(x), what can you say about lim x->0 tan(x)/x?

 

[Torshi]

tan(x)^2/x=(sin(x)/x)*(sin(x)/cos(x)^2). What's the limit of each of those two factors?

That would be 1 X 0 = 0 ?but I don't understand how tan(x)^2/x = what you posted above. .

 

[Torshi]

That had better be lim x-->0 (1-cosx)/x = 0 .

Do you know what cos(0) is? Given that, and knowing that sin(x)/x -> 0, and sin(x) = cos(x)tan(x), what can you say about lim x->0 tan(x)/x?

cos(0) = 1?

 

[Dick]

That would be 1 X 0 = 0


but I don't understand how tan(x)^2/x = what you posted above. .

tan(x)=sin(x)/cos(x). So tan(x)^2/x=sin(x)^2/(cos(x)^2*x)=(sin(x)/x)*(sin(x)/cos(x)^2). Just regroup the factors.

 

[SammyS]

... I know tanx = sinx/cosx.

Right.

So if $\displaystyle \ \ \tan(x)=\frac{\sin(x)}{\cos(x)}\,, \$ then $\displaystyle \ \ \frac{\tan^2(x)}{x}=\frac{\sin(x)\tan(x)}{x\cos(x)}\ . \$ Right?

Of course that can be written $\displaystyle \ \ \frac{\sin(x)}{x}\frac{\tan(x)}{\cos(x)}\ . \$

Does that help ?

 

[Torshi]

tan(x)=sin(x)/cos(x). So tan(x)^2/x=sin(x)^2/(cos(x)^2*x)=(sin(x)/x)*(sin(x)/cos(x)^2). Just regroup the factors.

Thank god I get it now! Jesus lol

 

[Torshi]

Right.

So if $\displaystyle \ \ \tan(x)=\frac{\sin(x)}{\cos(x)}\,, \$ then $\displaystyle \ \ \frac{\tan^2(x)}{x}=\frac{\sin(x)\tan(x)}{x\cos(x)}\ . \$ Right?

Of course that can be written $\displaystyle \ \ \frac{\sin(x)}{x}\frac{\tan(x)}{\cos(x)}\ . \$

Does that help ?

Yes! OMg

 

[Torshi]

Thank you!

 

[HallsofIvy]

Actually, just before SammyS's post Dick had told you that
$$\frac{tan^2 x}{x}= \frac{sin(x)}{x}\frac{1}{cos(x)}\frac{sin(x)}{cos(x)}$$
which, I think, is a simpler approach.