Ensino Superior ⇒ Limite Exponencial Fundamental Tópico resolvido

[aprendiz123]

[tex3]\lim_{x \to \infty} x[\ln(x+1) - \ln(x)][/tex3]

Resposta:

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1

[Karl Weierstrass]

[tex3]\lim_{x \to \infty}\,\, x\,\cdot\,[\ell n(x\,+\,1) \,- \,\ell n(x)][/tex3]

[tex3]\hspace{70pt}\lim_{x \to \infty} \,\,x\,\cdot\,[\ell n(x\,+\,1) \,- \,\ell n(x)]\,=\,\lim_{x \to \infty}\,\,x\,\cdot\,\ell n\, \frac{x\,+\,1}{x} [/tex3]

[tex3]\hspace{299pt}=\,\lim_{x \to \infty} \,\,\ell n\,\left( \frac{x\,+\,1}{x} \right)^x[/tex3]

[tex3]\hspace{299pt}=\,\ell n\,\underbrace{\lim_{x \to \infty} \,\,\left(1\,+\, \frac{1}{x} \right)^x}_{ \,\, e}[/tex3]

[tex3]\hspace{299pt}=\,\ell n\,e[/tex3]

[tex3]\hspace{299pt}=\,1.[/tex3]


[tex3](*)\,\, \log\, \frac{a}{b} \,=\,\log\,a\,-\,\log\,b[/tex3]

[tex3](**) \,\,\log\, a^c\,=\,c\,\cdot\,\log\,a[/tex3]





[tex3]\,[/tex3]

[aprendiz123]

vlw, tinha me esquecido das propriedades dos logaritmos