lim_{x→0+} (1+3x)^(1/x) = eł How do I fill in the blanks? In other words, how do I show that they are equal? Thank you!
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If $\displaystyle L=\lim_{x\to a}f(x)^{g(x)}$ presents an indetermination of the form $\displaystyle 1^{\infty}$ then, by a well known result $\displaystyle L=e^{\lambda}$ being $\displaystyle \lambda=\lim_{x\to a}(f(x)-1)g(x)$ .
Thank you! I'm pretty sure I recognize those results by using them in differential equations.
Originally Posted by CountingPenguins lim_{x→0+} (1+3x)^(1/x) = eł How do I fill in the blanks? In other words, how do I show that they are equal? Thank you! A well known definition of e is $\displaystyle \lim_{t \to +\infty} \left(1 + \frac{1}{t}\right)^t$. So make the substitution $\displaystyle 3x = \frac{1}{t}$ in your limit ....
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