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sara_87
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Homework Statement
Calculate the limit as x tends to 0:
(\frac{sinx}{x})^{\frac{1}{x^{2}}}
Limit of (sinx/x)^1/x^2 using L'Hospital's Rule?
- Thread starter sara_87
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SUMMARY
The limit of \((\frac{\sin x}{x})^{\frac{1}{x^{2}}}\) as \(x\) approaches 0 is evaluated using L'Hospital's Rule. As \(x\) approaches 0, \(\frac{\sin x}{x}\) approaches 1, leading to the expression simplifying to \(1^{\infty}\). Applying L'Hospital's Rule correctly reveals that the limit converges to \(e^{-1/2}\). This conclusion is definitive and relies on the application of calculus principles.
PREREQUISITES- Understanding of limits in calculus
- Familiarity with L'Hospital's Rule
- Knowledge of exponential functions
- Basic trigonometric identities
- Study the application of L'Hospital's Rule in various limit problems
- Explore the properties of exponential functions and their limits
- Learn about Taylor series expansions for trigonometric functions
- Investigate other forms of indeterminate limits
Students studying calculus, particularly those focusing on limits and L'Hospital's Rule, as well as educators seeking to clarify these concepts in teaching materials.
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