Question

lim x->0 ( 4x/tan(x) )

Please explain how to solve this?

Answer 1

You know of the identity
\[\lim_{x \rightarrow 0}\frac{ sinx }{ x }=1\]
correct?

Answer 2

Yes.

Answer 3

Well, thats exactly what were going to try to make happen. So considering I need sinx to do that, Ill change tanx first:

\[\frac{ 4x }{ \frac{ sinx }{ cosx } }= \frac{ \frac{ 4x }{ 1 } }{ \frac{ sinx }{ cosx } }\]
Now of course stacking it in this way isnt necessary, but it makes the visual better so ou can see what Im doing. So that sinx needs an x underneath it so I can make it go to 0. In order to get an x underneath it but still keep the equation the same, I multiply top and bottom by 1/x

\[\frac{ \frac{ 4x }{ 1 } }{ \frac{ sinx }{ cosx } }*\frac{ \frac{ 1 }{ x } }{ \frac{ 1 }{ x } }\]
Now because I managed to get a sinx/x situtation, I can say that whole portion is 1 if x goes to 0, so that leaves me with just:

\[\frac{ \frac{ 4x }{ x } }{ \frac{ 1 }{ cosx } }=4cosx\]Now if you plug in 0 for x you just get 4(1) = 4

Answer 4

Thank you so much for showing all the work to help me understand. I really appreciate it!
I also have another question about limits. Would you be willing to help me out with it?

Answer 5

What is it?

Answer 6

I have no idea what it is asking or how to solve it

Answer 7

The x>1 and x greater than or equal to 1 is what really confuses me , in addition to the fact that I don't know how to find a limit for two equations together like that.

Answer 8

Well, as you can see, there are two domains. One domain says x < 1 and the other x >= 1. So if limit is appraching 1, we can only pick one of the two domains. So the function that includes 1 in the domain is the one that has x >= 1, which is f(x) = -0.5x + 1. Kinda make sense why I choose that?

Answer 9

Ok. That makes more sense.
So use the function f(x)= -0.5x+1 and find the limit as x approaches 1
(-0.5)(1)+1= 0.5 ?

Answer 10

Yep.

Answer 11

Again, thank you for helping me!

Answer 12

yeah, np.