SUMMARY
The limit of the expression \(\lim_{x\to\frac{\pi}{2}}(x-\frac{\pi}{2})\tan x\) evaluates to -1. The solution involves rewriting the tangent function using the identity \(\tan x = \frac{1 - \cos(2x)}{\sin(2x)}\), which simplifies the limit calculation. This approach effectively transforms the limit into a more manageable form, allowing for straightforward evaluation as \(x\) approaches \(\frac{\pi}{2}\).
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with trigonometric identities
- Knowledge of L'Hôpital's Rule
- Experience with rewriting expressions for limit evaluation
NEXT STEPS
- Study L'Hôpital's Rule for evaluating indeterminate forms
- Explore trigonometric identities and their applications in limits
- Learn about Taylor series expansions for trigonometric functions
- Investigate advanced limit techniques in calculus
USEFUL FOR
Students studying calculus, particularly those focusing on limits and trigonometric functions, as well as educators seeking effective methods for teaching these concepts.