Evaluate the Limit limit as x approaches pi of cot(x)

Problem

(lim_x→π)(cot(x))

Solution

1. Rewrite the expression using the definition of the cotangent function in terms of sine and cosine.

cot(x)=cos(x)/sin(x)

2. Evaluate the limits of the numerator and the denominator separately as x approaches π

(lim_x→π)(cos(x))=cos(π)=−1

(lim_x→π)(sin(x))=sin(π)=0

3. Analyze the behavior of the fraction as the denominator approaches zero while the numerator remains a non-zero constant.

(lim_x→π)((−1)/0)

4. Determine the one-sided limits to check for existence. As x→π− sin(x) is positive, so the limit is −∞ As x→π+ sin(x) is negative, so the limit is +∞

(lim_x→π−)(cot(x))=−∞

(lim_x→π+)(cot(x))=∞

5. Conclude that because the one-sided limits do not match and the function grows without bound, the limit does not exist.

Final Answer

(lim_x→π)(cot(x))=Does Not Exist

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