Math Problem Statement

lim(x->(pi-/2)^-) acosx/(pi - 2x)

Solution

To evaluate the given limit:

limx(π2)arccosxπ2x,\lim_{x \to \left(\frac{\pi}{2}\right)^-} \frac{\arccos x}{\pi - 2x},

let us proceed step by step.


Step 1: Analyze the behavior of the numerator (arccosx\arccos x)

The arccosx\arccos x function approaches arccos(cosπ2)=π2\arccos\left(\cos\frac{\pi}{2}\right) = \frac{\pi}{2} as xx approaches 1 (from the left). However, as xπ2x \to \frac{\pi}{2}^-, the exact functional behavior will depend on substitution for clarity.

Step 2 Above]]) **Tip code samples

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometric Functions
Continuity

Formulas

lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x) (L'Hôpital's Rule)
Inverse Trigonometric Functions

Theorems

L'Hôpital's Rule
Behavior of Inverse Trigonometric Functions near their bounds

Suitable Grade Level

Undergraduate Mathematics