Evaluate the Limit limit as x approaches 0 of sec(x)

Problem

(lim_x→0)(sec(x))

Solution

1. Identify the function and the point of evaluation. The function is sec(x) and the limit is as x approaches 0

2. Recall the definition of the secant function in terms of the cosine function.

sec(x)=1/cos(x)

3. Check for continuity at the point x=0 Since cos(0)=1 which is non-zero, the function sec(x) is continuous at x=0

4. Apply the direct substitution property for limits of continuous functions.

(lim_x→0)(sec(x))=sec(0)

5. Evaluate the trigonometric value.

sec(0)=1/cos(0)=1/1=1

Final Answer

(lim_x→0)(sec(x))=1

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