Why Does (sin(2x))/x Equal 2 as x Approaches 0?

[hahaha158]

Homework Statement

I have the equation (sin(2x))/x = ?

The Attempt at a Solution

I know that the answer to this is 2, but I am not sure why (sin(2x))/x = 2

Can someone explain?

Thanks

 

[Mark44]

Homework Statement

I have the equation (sin(2x))/x = ?

The Attempt at a Solution

I know that the answer to this is 2, but I am not sure why (sin(2x))/x = 2

Can someone explain?

(sin(2x))/x ≠ 2, so perhaps you are leaving something out of the problem. What is the complete problem statement?

 

[hahaha158]

(sin(2x))/x ≠ 2, so perhaps you are leaving something out of the problem. What is the complete problem statement?

Find the value of the constant a for which the function below is continuous everywhere. Fully
explain your reasoning.

... a+x² while x≤0
f(x) = {
... (sin(2x))/x while x>0

 

[Mark44]

The problem is really asking about limits, namely
$$ \lim_{x \to 0}\frac{sin(2x)}{x}$$

Do you know any other limits that involve trig functions?

 

[hahaha158]

The problem is really asking about limits, namely
$$ \lim_{x \to 0}\frac{sin(2x)}{x}$$

Do you know any other limits that involve trig functions?

I'm not sure what you mean by your question.

I know that $$ \lim_{x \to 0}\frac{sin(x)}{x}$$ is equal to 1

I also know that the limit as x approaches 0 from the negative and the limit at x=0 are both just equal to a.

So this means that a just equals $$ \lim_{x \to 0}\frac{sin(2x)}{x}$$

I know the answer to the question is 2 so that means $$ \lim_{x \to 0}\frac{sin(2x)}{x}$$ must equal 2 but I am not sure how to do it.

 

[Mark44]

There are at least a couple of ways to go.
1) Double angle identity for sine
2) Adjust things so that you have sin(2x)/(2x) times some other stuff.

 

[hahaha158]

There are at least a couple of ways to go.
1) Double angle identity for sine
2) Adjust things so that you have sin(2x)/(2x) times some other stuff.

For 2) do you mean like

(sin(2x))/2x)*2

=so you get 1*2

=2?

Would that work?

 

[5ymmetrica1]

(sin(2x))/ x = (sin2(1))/ 1
= sin(2)(1)/1
= sin(2)
= 0.0349

 

[Mentallic]

(sin(2x))/ x = (sin2(1))/ 1
= sin(2)(1)/1
= sin(2)
= 0.0349

That's the limit as x goes to 1, not 0.

For 2) do you mean like

(sin(2x))/2x)*2

=so you get 1*2

=2?

Would that work?

Yep, exactly! This would be the simplest way to do it, so if you ever get a question like

$$\lim_{x\to 0}\frac{\sin(ax)}{b}$$ then this is equivalent to
$$\lim_{x\to 0}\frac{a}{b}\cdot\frac{\sin(ax)}{a}=\frac{a}{b}$$