Determine the following limits.

Question

\lim_{x \to 0^{-}} \csc (x)

Accepted Answer

Welcome back, everyone. In this problem, we want to find the limit as X approaches 0 of -2 multiplied by the coc count of 5 halves of X. A says it's infinity. B says it's 0, C negative infinity, and the D says it does not exist. Now, in order to find the limit, let's first make sure we understand the function, OK? So our given function is negative2 multiplied by the cocant of 5 halves of X, and essentially we're looking for the behavior of this function as X approaches 0 from the left, OK. Now, what do we know about the coreC can function? Well, recall, OK, that the cosicant of X is the reciprocal of the sine function. So it's going to be one divided by the sine of X. So with that in mind, then that means we can rewrite -2 multiplied by the core scant of 5 halves of X. As 2 multiplied by 1 divided by the sine of 5 halves of X. Or -2 divided by the sign of 5 halves of X. Now what do we know then about the sign of the function and the behavior of what's happening inside the sign function that is 5 halves of X. Well, let's analyze the behavior for our limit. So know, to find the limit as x approaches 0 from the left of -2. Divided by the sign of 5 houses of X, we're going to ask ourselves what happens to the limit of the sign function and the term inside the sign function. Now starting with the term inside the sign function, as X approaches 0 from the left, OK, then 5 halves of X. It is also going to approach 0 from the left. In other words, it's going to give us a small negative value. Note as well that the sign of a small negative value is also a small negative value using what we know about small angle approximation. So this also tells us then. That the sign OK, uh, 5 halves of Xs. Approaches the sign of 0 from the left, which based on small approximation, is also going to approach a small negative value from the left. So what does that tells us about our limit then is that the limit as X approaches 0 from the left of our expression. Is going to be equal to -2 divided by 0 being approached from the left a small negative value and -2 divided by a small negative value is going to be equal to positive infinity. Therefore, the limit equals infinity. If we look back on our answer choices, that tells us a is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine the following limits.

$\lim_{x \to 0^{-}} \csc (x)$

Key Concepts

Limits

Cosecant Function

One-Sided Limits

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