1) My last question was wrong. The question should read: Fill in the blanks

As x --> pi+, sin x --> ?

and csc x --> ?

I do not understand this!

sin (PI)=0

so what is csc(x) as x-->0

oops ,as x-->pi

No problem! I'm here to help you understand.

When we say "as x approaches pi+," it means that x is getting closer and closer to pi from the positive (right) side. In this context, we need to determine the limit of sin(x) and csc(x) as x approaches pi from the positive side.

To find the limit as x approaches pi+ for sin(x), we can evaluate the sine function at progressively smaller values of x that are getting closer to pi from the right side. We know that the sine function oscillates between -1 and 1 as x varies. As x approaches pi from the positive side, sin(x) will approach 0. Therefore, as x --> pi+, sin(x) --> 0.

Now, let's determine the limit as x approaches pi+ for csc(x). The csc(x) function represents the reciprocal of the sine function (1/sin(x)). Since we know that sin(x) approaches 0 as x approaches pi from the positive side, the reciprocal (1/0) is undefined (or tends towards infinity). Therefore, as x --> pi+, csc(x) is undefined or tends towards infinity.

So, for the given question:

As x --> pi+, sin(x) --> 0
And csc(x) is undefined or tends towards infinity.