Evaluate the Limit

Problem

(lim_x→π/2)(sec(x)−tan(x))

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine to identify the indeterminate form.

sec(x)−tan(x)=1/cos(x)−sin(x)/cos(x)

2. Combine the terms into a single fraction with a common denominator.

(1−sin(x))/cos(x)

3. Evaluate the limit at x=π/2 to check for an indeterminate form.

(1−sin(π/2))/cos(π/2)=(1−1)/0=0/0

4. Apply L'Hôpital's Rule by differentiating the numerator and the denominator separately.

d(1−sin(x))/d(x)=−cos(x)

d(cos(x))/d(x)=−sin(x)

5. Simplify the resulting expression after differentiation.

(−cos(x))/(−sin(x))=cos(x)/sin(x)

6. Substitute x=π/2 into the simplified expression to find the limit.

cos(π/2)/sin(π/2)=0/1=0

Final Answer

(lim_x→π/2)(sec(x)−tan(x))=0

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