Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
91. lim(x→π/2⁻) (sec(7x))(cos(3x))

Verified step by step guidance

1. First, rewrite the limit expression to understand its behavior as \(x\) approaches \(\frac{\pi}{2}\) from the left: \(\lim_{x \to \frac{\pi}{2}^-} (\sec(7x)) (\cos(3x))\).
2. Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), so rewrite the expression as \(\lim_{x \to \frac{\pi}{2}^-} \frac{\cos(3x)}{\cos(7x)}\).
3. Evaluate the numerator and denominator separately at \(x = \frac{\pi}{2}\) to check if the limit is an indeterminate form: \(\cos(3 \cdot \frac{\pi}{2})\) and \(\cos(7 \cdot \frac{\pi}{2})\).
4. If direct substitution results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), apply l’Hôpital’s Rule by differentiating the numerator and denominator with respect to \(x\):
5. \[\lim_{x \to \frac{\pi}{2}^-} \frac{\frac{d}{dx} \cos(3x)}{\frac{d}{dx} \cos(7x)} = \lim_{x \to \frac{\pi}{2}^-} \frac{-3 \sin(3x)}{-7 \sin(7x)} = \lim_{x \to \frac{\pi}{2}^-} \frac{3 \sin(3x)}{7 \sin(7x)}.\]
6. Finally, evaluate this new limit by substituting \(x = \frac{\pi}{2}\) and simplify to find the limit.

Key Concepts

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.

Trigonometric Limits and Continuity

Understanding the behavior of trigonometric functions near specific points is essential. Knowing how functions like secant and cosine behave as their arguments approach values like π/2 helps in simplifying and evaluating limits.

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a point from only one side (left or right). This is important when the function behaves differently on either side, such as near vertical asymptotes or discontinuities.