Limit of [1 + sin(x)]^(1/x) when x approaches 0

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SUMMARY

The limit of [1 + sin(x)]^(1/x) as x approaches 0 can be effectively solved using logarithmic transformation and L'Hôpital's Rule. By rewriting the expression as (1 + sin(x))^(1/sin(x))^(sin(x)/x), the limit simplifies, allowing for straightforward evaluation. The correct limit is determined to be e, confirming the application of these mathematical techniques yields accurate results.

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  • Knowledge of logarithmic functions
  • Basic trigonometric functions, specifically sin(x)
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Can somebody solve this problem: the limit of [1 + sin(x)]^(1/x) when x approaches 0 ?
 
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Take the logarithm of that expression, which will let you use L'Hopital to solve it.
 
The simplest is to rewrite this as:
[tex](1+\sin(x))^{\frac{1}{x}}=((1+\sin(x))^{\frac{1}{\sin(x)}})^{\frac{\sin(x)}{x}}[/tex]
The correct limit is quite easy to deduce from this.
 

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