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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 negative

values as $x \rightarrow 2 \pi^{-}$

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we value this limit. He knows that Costa rica index. This is equal to one over Synnex. And so we can write this as limit as X approaches to pie from the left of X times one over cynics. And using limit loss, we can write this as limit as X approaches to buy from the left of X times the limit as X approaches to buy from the left Of one over Synnex. Now evaluating at two by we have to buy times. Yeah, one over sine of two by In which sign of to buy approaches zero. Now since X approaches to buy from the left, we know that X is in the fourth quadrant where sign is negative. So the value of sign of two pi is a small negative number. And if that's the case the value of one over sine of two, but I will be a negative infinite number. And so in here we have to buy times negative infinity. This will give us negative infinity. And so this is the limit of the function