Evaluate the Limit

Problem

(lim_x→π/2)(cos(x)/(1−sin(x)))

Solution

1. Identify the form of the limit by substituting x=π/2 into the expression.

2. Evaluate the numerator and denominator: cos(π/2)=0 and 1−sin(π/2)=1−1=0

3. Recognize that the limit results in the indeterminate form 0/0 which allows the use of L'Hôpital's Rule.

4. Apply L'Hôpital's Rule by taking the derivative of the numerator and the derivative of the denominator separately.

5. Differentiate the numerator: d(cos(x))/d(x)=−sin(x)

6. Differentiate the denominator: d(1−sin(x))/d(x)=−cos(x)

7. Simplify the new expression: (−sin(x))/(−cos(x))=tan(x)

8. Evaluate the limit of the new expression as x approaches π/2 from the left: (lim_x→π/2−)(tan(x))=∞

9. Evaluate the limit of the new expression as x approaches π/2 from the right: (lim_x→π/2+)(tan(x))=−∞

10. Conclude that because the one-sided limits are not equal (and specifically, the function grows without bound), the limit does not exist.

Final Answer

(lim_x→π/2)(cos(x)/(1−sin(x)))=Does Not Exist

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