Evaluate the Limit limit as x approaches 0 of csc(x)

Problem

(lim_x→0)(csc(x))

Solution

1. Rewrite the cosecant function in terms of the sine function using the reciprocal identity.

csc(x)=1/sin(x)

2. Analyze the behavior of the sine function as x approaches 0 from the right.

(lim_x→0)(sin(x))=0

3. Determine the right-hand limit by observing that the reciprocal of a very small positive number is a very large positive number.

(lim_x→0)(1/sin(x))=∞

4. Analyze the behavior of the sine function as x approaches 0 from the left.

(lim_x→0)(sin(x))=0

5. Determine the left-hand limit by observing that the reciprocal of a very small negative number is a very large negative number.

(lim_x→0)(1/sin(x))=−∞

6. Compare the one-sided limits. Since the left-hand limit and the right-hand limit are not equal, the two-sided limit does not exist.

∞≠−∞

Final Answer

(lim_x→0)(csc(x))=Does Not Exist

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