Evaluate Using L'Hospital's Rule

Problem

(lim_x→π/2)(cos(x)/(1−sin(x)))

Solution

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

cos(π/2)=0

1−sin(π/2)=1−1=0

The limit is in the indeterminate form 0/0

2. Apply L'Hospital's Rule by differentiating the numerator and the denominator separately.

d(cos(x))/d(x)=−sin(x)

d(1−sin(x))/d(x)=−cos(x)

3. Rewrite the limit using the derivatives found in the previous step.

(lim_x→π/2)((−sin(x))/(−cos(x)))

4. Simplify the expression before evaluating.

(−sin(x))/(−cos(x))=tan(x)

5. Evaluate the limit as x approaches π/2 from the left, as the function is undefined at π/2 and the denominator approaches zero.

(lim_x→π/2−)(tan(x))=∞

6. Re-evaluate the original limit expression behavior. As x→π/2 sin(x)→1 and cos(x)→0 Specifically, for x near π/2 cos(x) and 1−sin(x) are both positive, leading to positive infinity.

Final Answer

(lim_x→π/2)(cos(x)/(1−sin(x)))=∞

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