Use l’Hôpital’s rule to find the limits in Exercises 7–52.

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

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First, identify the form of the limit as \( \theta \to \frac{\pi}{2} \). Substitute \( \theta = \frac{\pi}{2} \) into the expression \( \frac{2\theta - \pi}{\cos(2\pi - \theta)} \) to check if it results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

Since direct substitution gives \( 2\left(\frac{\pi}{2}\right) - \pi = 0 \) and \( \cos(2\pi - \frac{\pi}{2}) = \cos\left(\frac{3\pi}{2}\right) = 0 \), the limit is of the indeterminate form \( \frac{0}{0} \), so l’Hôpital’s Rule applies.

Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \( \theta \). The derivative of the numerator \( 2\theta - \pi \) is \( 2 \).

The derivative of the denominator \( \cos(2\pi - \theta) \) requires the chain rule: \( \frac{d}{d\theta} \cos(2\pi - \theta) = -\sin(2\pi - \theta) \times (-1) = \sin(2\pi - \theta) \).

Rewrite the limit as \( \lim_{\theta \to \frac{\pi}{2}} \frac{2}{\sin(2\pi - \theta)} \) and then substitute \( \theta = \frac{\pi}{2} \) to evaluate the limit.

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