Limits of log sin equal to 0/1/infinity?

[LASmith]

Homework Statement

lim_x->$\pi$/2(ln sin x)/(($\pi$-2x)²)

Homework Equations

The Attempt at a Solution

Putting $\pi/2$ into this equation gives 0/0, however, the limit of this could be 1,0 or infinity, so how would you distinguish between the three?

 

[tiny-tim]

Hi LASmith!

l'Hôpital's rule? ​

 

[uart]

Putting $\pi/2$ into this equation gives 0/0, however, the limit of this could be 1,0 or infinity, so how would you distinguish between the three?

LASmith, note that it doesn't have to be one of those three options.

 

[NascentOxygen]

Homework Statement

lim_x->$\pi$/2(ln sin x)/(($\pi$-2x)²)

This being precalculus, I think it would be sufficient to evaluate the function for a few values very close to Pi/2, on one side then the other. For example, Pi/2 - 0.001, Pi/2 - 0.002, Pi/2 - 0.005 and also try Pi/2 + 0.001 etc. (Then graph the points, at least in your mind, so you can see what the function is doing in the vicinity of Pi/2.)

Ideally, you might think it would be useful to evaluate Pi/2 - 0.00000000001 etc. so as to be really close to the point of interest, Pi/2, but this many sig figs will exceed the capabilities of your calculator. There is software which will implement a calculator that can work accurately to hundreds of decimal places, but we really don't need such extravagance here. (Though it is fun to experiment with to confirm what happens really really close to difficult points. )

 

[PeterO]

Homework Statement

lim_x->$\pi$/2(ln sin x)/(($\pi$-2x)²)

Homework Equations

The Attempt at a Solution

Putting $\pi/2$ into this equation gives 0/0, however, the limit of this could be 1,0 or infinity, so how would you distinguish between the three?

That is why we have lim_x->$\pi$/2

We consider what happens when x gets very close to $\pi$/2 [from above and below] but never actually equaling $\pi$/2.

 

[gat0man]

Consider: What are both of the functions doing as X-> pi/2?

As you are in precalc and not expected to use L'Hopitals Rule, look at the graphs of these functions and consider which has larger values as X-> pi/2

If you have a very small value in the numerator and a larger value in the denominator, what happens?