Evaluate the Limit

Problem

(lim_x→π/2)((−2)/cos(x))

Solution

1. Analyze the behavior of the denominator as x approaches π/2

2. Evaluate the limit of the denominator: (lim_x→π/2)(cos(x))=0

3. Determine the behavior of the fraction as the denominator approaches zero while the numerator remains a constant −2

4. Consider the one-sided limits because cos(x) changes sign at π/2

5. Evaluate the left-hand limit: as x→π/2− cos(x) is positive and approaching 0 so (−2)/cos(x)→−∞

6. Evaluate the right-hand limit: as x→π/2+ cos(x) is negative and approaching 0 so (−2)/cos(x)→∞

7. Conclude that since the one-sided limits are not equal, the two-sided limit does not exist.

Final Answer

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

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