Ensino Superior ⇒ Limite Exponencial Fundamental

[aprendiz123]

[tex3]\lim_{n \to \infty} \left(\frac{2n+3}{2n+1}\right)^{1+n}[/tex3]

Resposta:

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[tex3]e[/tex3]

[Karl Weierstrass]

• [tex3]\begin{array}{rl}
  \lim_{n \to \infty}\text{ } \left(\Large\frac{2n\,+\,3}{2n\,+\,1}\large\right)^{1\,+\,n}& = \lim_{n \to \infty}\text{ } \left(\Large\frac{2n\,+\,1\,+\,2}{2n\,+\,1}\large\right)^{1\,+\,n}\\
  &=\lim_{n \to \infty}\text{ } \left(1\,+\,\Large\frac{2}{2n\,+\,1}\large\right)^{1\,+\,n} \\
  &=\lim_{n \to \infty}\text{ } \left(1\,+\,\Large\frac{1}{n\,+\,\Large\frac{1}{2}\large}\large\right)^{n\,+\,\Large\frac{1}{2}\large\,+\,\Large\frac{1}{2}\large} \\
  &=\underbrace{\lim_{n \to \infty}\text{ } \left(1\,+\,\Large\frac{1}{n\,+\,\Large\frac{1}{2}\large}\large\right)^{n\,+\,\Large\frac{1}{2}\large}}_{\large \text{ }e}\,\cdot\,\underbrace{\lim_{n \to \infty}\text{ } \left(1\,+\,\Large\frac{1}{n\,+\,\Large\frac{1}{2}\large}\large\right)^{\Large\frac{1}{2}\large}}_1 \\
  &=e.
  \end{array}[/tex3]