Evaluate the Limit limit as x approaches pi/2 of xsec(x)^2

Problem

(lim_x→π/2)(x)*sec2(x)

Solution

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

2. Evaluate the behavior of the trigonometric function sec(x) as x approaches π/2

3. Observe that sec(x)=1/cos(x) As x→π/2 cos(x)→0 which means sec(x)→±∞

4. Determine the behavior of the squared term sec2(x) Since any real number squared is non-negative, sec2(x)→∞ as x→π/2

5. Analyze the product x*sec2(x) As x→π/2 the first factor x approaches the positive constant π/2 and the second factor sec2(x) approaches ∞

6. Conclude that the product of a positive constant and a value approaching infinity results in the limit approaching infinity.

Final Answer

(lim_x→π/2)(x)*sec2(x)=∞

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