Use l’Hôpital’s rule to find the limits in Exercises 7–52.
27. lim (x → (π/2)^-) (x - π/2) sec x

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Identify the limit expression: \(\lim_{x \to (\pi/2)^-} (x - \pi/2) \sec x\).

Check the form of the limit by substituting \(x = \pi/2\) from the left side: \((x - \pi/2)\) approaches 0, and \(\sec x = \frac{1}{\cos x}\) tends to \(\pm \infty\) because \(\cos(\pi/2) = 0\). This suggests an indeterminate form of type \(0 \cdot \infty\).

Rewrite the expression to apply l’Hôpital’s rule by converting the product into a quotient. For example, write it as \(\frac{x - \pi/2}{\cos x}\) because \(\sec x = \frac{1}{\cos x}\).

Now the limit becomes \(\lim_{x \to (\pi/2)^-} \frac{x - \pi/2}{\cos x}\), which is of the form \(\frac{0}{0}\), suitable for l’Hôpital’s rule.

Apply l’Hôpital’s rule by differentiating numerator and denominator separately: differentiate numerator \(\frac{d}{dx}(x - \pi/2) = 1\), and denominator \(\frac{d}{dx}(\cos x) = -\sin x\). Then evaluate the new limit \(\lim_{x \to (\pi/2)^-} \frac{1}{-\sin x}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits and One-Sided Limits

A limit describes the value a function approaches as the input approaches a certain point. One-sided limits consider the approach from only one side, such as from the left (denoted by the minus sign). Understanding how to evaluate these limits is essential for analyzing behavior near points where the function may be undefined.

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met. This rule simplifies complex limit problems.

Behavior of Trigonometric Functions Near Critical Points

Trigonometric functions like sec(x) can have vertical asymptotes or undefined points at specific values, such as π/2 for sec(x). Understanding how these functions behave near such points helps in correctly applying limit techniques and interpreting the function’s behavior as x approaches these critical values.

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