Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 
Try it—you just keep on cycling. Find the limits some other way.
71. lim (x → (π/2)⁻) sec x / tan x

Verified step by step guidance

1. Identify the limit expression: \(\lim_{x \to (\pi/2)^-} \frac{\sec x}{\tan x}\).
2. Rewrite the trigonometric functions in terms of sine and cosine to simplify the expression: \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\). Substitute these into the limit to get \(\lim_{x \to (\pi/2)^-} \frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}}\).
3. Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}} = \frac{1}{\cos x} \times \frac{\cos x}{\sin x} = \frac{1}{\sin x}\).
4. Now the limit reduces to \(\lim_{x \to (\pi/2)^-} \frac{1}{\sin x}\). Consider the behavior of \(\sin x\) as \(x\) approaches \(\pi/2\) from the left side.
5. Since \(\sin x\) approaches 1 as \(x\) approaches \(\pi/2\), analyze the limit \(\frac{1}{\sin x}\) accordingly to determine the limit value.

Key Concepts

Behavior of Trigonometric Functions Near Specific Points

Understanding how functions like sec x and tan x behave as x approaches π/2 from the left is crucial. Near π/2, tan x tends to infinity, while sec x also grows large but their rates differ, affecting the limit's value.

Limits Involving Indeterminate Forms and Alternative Techniques

When L’Hôpital’s Rule leads to cyclic or unresolvable forms, alternative methods such as algebraic manipulation, trigonometric identities, or analyzing one-sided limits must be used to evaluate the limit.

One-Sided Limits and Directional Approach

Evaluating limits as x approaches π/2 from the left (denoted π/2⁻) requires considering the function's behavior just before π/2, which can differ from approaching from the right, impacting the limit's existence and value.