Vertical Asymptote of f(x) = 2x/sin2x

[gator]

Question: Find the Vertical Asymptote if any

f(x)=[tex}\frac{2x}{\sin2x}[\tex]

Im somewhat lost, I cross multi and get 1/2 although my friends tell me this just equals 1. Any ideas?

err...cant get the cool math symbol things to work...

 

[TD]

Could you clarify the function? The LaTeX didn't work properly, check your tags.

 

[gator]

im trying to say f(x)= 2x / sin2x

 

[TD]

Use the correct tags (closing with /, not \) and click to see the source:

\frac{2x}{\sin2x}

Now, you can encounter a vertical asymptote if the denominator becomes zero for some value of x while the nominator is non-zero there.
If both are zero, you can check with a limit what the function becomes.

 

[gator]

The only thing that would have the denom zero would be zero (sin2(o)) = 0
but that would cause the nom to become zero

 

[TD]

Both become zero there indeed, in which case you can't tell (yet) if there will be a vertical asymptote. Have you covered limits?
You forgot one thing though, the sine-function becomes zero somewhere else (of course, it's period with period 2\pi, but it has another zero within [0,2\pi]). Think about the trigonometric unit circle, when is the sine zero as well?

 

[gator]

sin = zero @ pi (180* on unit circle)

yep, covered limits. But in our notes he covers finding the VA without using limits (weird i know) so I am not sure

 

[TD]

sin = zero @ pi (180* on unit circle)

yep, covered limits. But in our notes he covers finding the VA without using limits (weird i know) so I am not sure

Indeed, at x = \pi as well! And the nominator doesn't become zero there, so we have a real vertical asymptote here.
Of course, don't forget the sin(x) is periodic, so not only at x = \pi but also at...? And the nominator only became zero at x = 0, but the denominator became zero there, so also at multiples of 2 \pi where the nominator doesn't become zero anymore!
Can you follow?

Perhaps you saw a standard limit you may use for the 0/0 case, which is the well-known:

\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = 1

 

[gator]

So my VA is x=pi ?

 

[TD]

So my VA is x=pi ?

There's one yes, but there are more. Reread my last reply, I edited while you were posting.

 

[gator]

sin2x is zero when x=1/2 pi ? since that would mean sin = pi which would be zero on the denom

 

[TD]

Well, sin(x) is a periodic function with period 2\pi. This means that if it becomes zero at x = 0, it also becomes zero when you add a multiple of 2\pi. The same goes for the case x = \pi.

So we have that sin(x) becomes zero at: x = 0 + 2k\pi \vee x = \pi + 2k\pi where k is an integer. Now, the nominator is non-zero in all those cases, except for x = 0 (but not for its multiples of 2\pi). So if we leave out x = 0 in the x values I listed, then we have all the places where a vertical asymptote occurs. You see?

 

[gator]

ok, i have you until the last sentsnce. if "k" is an interger, then the amount of VA's i could have is endless.

Im also a little lost on what i should write down on my assignment

by the way, thanks for all this help

 

[TD]

Well, you're right! This function has an infinite number of vertical asymptotes!

 

[gator]

lol yea!

now are you any good at continuity? I have a question

using the def. of continuity to determine the points at which the function is no continuous and indicate whether the discontinuity is removable or not

f(x) = HUGE bracket (x+1)absolute value x-2 all over 2 sqrt x+1 x<0
1 x=0
-e^x / x+2 x>0

I have no notes on continuity, so if you could just lead me to a good site if you don't have time to help then that fine

 

[gator]

and a quick question on limits

lim x right arrow 0 e^3x -1 / e^x -1

since anything to the root 0 is 1 you get 1-1/1-1 = 0

 

[TD]

and a quick question on limits

lim x right arrow 0 e^3x -1 / e^x -1

since anything to the root 0 is 1 you get 1-1/1-1 = 0

So you're looking for

\mathop {\lim }\limits_{x \to 0} \frac{{e^{3x} - 1}}{{e^x - 1}}

Have you seen L'Hopital's rule? Or can you factor the nominator knowing that e^{3x} - 1 = \left( {e^x } \right)^3 - 1^3?

 

[gator]

L'Hopital's rule?

 

[TD]

Oh I see you made a mistake: (1-1)/(1-1) is 0/0 which isn't the same as 0...! You encounter an 'indeterminate form' here, namely 0/0 where you can apply L'Hopital's rule which states that in this case, you can take the nominator's and denominator's derivative (seperately, so f/g becomes f'/g'). If you don't know this rule, try factoring so you'll be able to cancel out e^x-1.

 

[gator]

so like (e^x - 1)(e^x2 -e^x +1) so I now have e^x2 - e^x +1

 

[TD]

so like (e^x - 1)(e^x2 -e^x +1) so I now have e^x2 - e^x +1

I think you made a small mistake, the second factor is (e^(2x) + e^x +1), it should have a plus. Then fill in x = 0 now