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Determine the infinite limit.

$ \displaystyle \lim_{x \to 2\pi^-}x\csc x $

$\lim _{x \rightarrow 2 \pi^{-}} x \csc x=\lim _{x \rightarrow 2 \pi-} \frac{x}{\sin x}=-\infty$ since the numerator is positive and the denominator approaches 0 through negativevalues as $x \rightarrow 2 \pi^{-}$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Limits

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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this is problem number thirty nine Stuart Calculus A edition section two point two. Determine the infinite limit. The limit has X approaches to pine from the left of the function X Times cause he can ex It's what value we're going to first, I think a slight modification to the limit we're going tow convert Cosi kin and to a function some more familiar to us. Kam usually b a very good first step in our system before the problem, using the definition of Corsican as one divided by sign, we've rewritten limit has the limited's X approaches to pine from the left of the function X times one over cynics. Now we wanted to know the behavior of sine x, a protein to pie from the left. So we use the rafter to our right as reference and as we follow the function along as we approach to pie from the left. For this function, we approached the value of zero. But most importantly, we have put a value is zero from a negative reference, and so we should has to meet the value of sign as a very small negative number. So number very close to zero very, very small and negative. Some something maybe. Like Negative Point Tero zero. There is where I want you take the X fire. Put it on top. In this case, it's too prying, regardless of the numerator is long as it is a party night number. Any pregnant, divided bravery, small number gives us an infinite value. And because this and this infinitely small value here was negative, means a hour ratio here is negative. Infinity to repeat. We condensed the function, too, a former sign and then we had it as a racial exit surrounded by cynics or cynics tends to be a very, very small negative number as X approaches to pie from the left, which gives us a negative infinity value since signs in the denominator. In order to confirm our findings, we plant the the graph of X Tim's cozy necks, Cosi connects, and as we approached two pi, which is approximately six point two eight, we see that be function tennis towards the negative infinity in this confirms our solution for this problem

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