Evaluate the Limit

Problem

(lim_x→π/2)((−2)/cos(x))

Solution

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

2. Evaluate the numerator, which is a constant −2

3. Evaluate the denominator cos(x) as x→π/2

4. Observe that cos(π/2)=0

5. Determine the direction of the limit. As x→π/2 from the left (x→π/2−, cos(x) is positive and approaches 0 so the fraction approaches −∞

6. Determine the direction of the limit. As x→π/2 from the right (x→π/2+, cos(x) is negative and approaches 0 so the fraction approaches +∞

7. Conclude that because the left-hand and right-hand 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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