Ensino Superior ⇒ Limite Exponencial Fundamental Tópico resolvido

[aprendiz123]

[tex3]\lim_{x \to \ 2} \frac{3^{\sqrt{x}} - 3^{\sqrt{2}}}{x-2}[/tex3]

Resposta:

---

[tex3]\frac{ 3^{\sqrt{x}}}{2\sqrt{2}}\ln3[/tex3]

[Karl Weierstrass]

[tex3]\lim_{x \to 2}\,\, \frac{3^{\sqrt{x}} \,-\, 3^{\sqrt{2}}}{x\,-\,2} [/tex3]

[tex3]\lim_{x \to 2}\,\, \frac{3^{\sqrt{x}} \,-\, 3^{\sqrt{2}}}{x\,-\,2} \,=\,\lim_{x \to 2}\,\, \frac{3^{\sqrt{2}}\,\cdot\,(3^{\sqrt{x}\,-\,\sqrt{2}} \,-\, 1)}{x\,-\,2} [/tex3]

[tex3]=\,\lim_{x \to 2}\,\, (3^{\sqrt{x}\,-\,\sqrt{2}} \,-\, 1)\cdot\, \frac{3^{\sqrt{2}}}{x\,-\,2} \,\cdot\, \frac{\sqrt{x}\,-\,\sqrt{2}}{\sqrt{x}\,-\,\sqrt{2}} \,\cdot\, \frac{\sqrt{x}\,+\,\sqrt{2}}{\sqrt{x}\,+\,\sqrt{2}} [/tex3]

[tex3]=\,\lim_{x \to 2}\,\, \frac{3^{\sqrt{x}\,-\,\sqrt{2}} \,-\, 1}{\sqrt{x}\,-\,\sqrt{2}} \,\cdot\, \frac{3^{\sqrt{2}}}{\sqrt{x}\,+\,\sqrt{2}} [/tex3]

[tex3]=\,\underbrace{\lim_{x \to 2}\,\, \frac{3^{\sqrt{x}\,-\,\sqrt{2}} \,-\, 1}{\sqrt{x}\,-\,\sqrt{2}} }_{ \ell n \,3}\,\cdot\,\lim_{x \to 2}\,\, \frac{3^{\sqrt{2}}}{\sqrt{x}\,+\,\sqrt{2}} [/tex3]

[tex3]=\, \frac{3^{\sqrt{2}}\,\cdot\,\ell n \,3}{2\sqrt{2}} [/tex3].


[tex3]\lim_{x \to 2}\,\, \frac{3^{\sqrt{x}\,-\,\sqrt{2}} \,-\, 1}{\sqrt{x}\,-\,\sqrt{2}} [/tex3]

[tex3]\left|\begin{array}{l}y\,=\,\sqrt{x}\,-\,\sqrt 2\\ x\to 2 \,\Longrightarrow\,y\to 0\end{array}\right.[/tex3]

[tex3]\lim_{x \to 2}\,\, \frac{3^{\sqrt{x}\,-\,\sqrt{2}} \,-\, 1}{\sqrt{x}\,-\,\sqrt{2}} \,=\,\lim_{y \to 0}\,\, \frac{3^y \,-\, 1}{y} \,=\,\ell n \,3[/tex3]


[tex3](*)\,\,\lim_{x \to 0}\, \frac{a^x \,-\,1}{x} \,=\,\ell n\, a[/tex3]


[tex3]\,[/tex3]

[aprendiz123]

grande, isso ae matou todos em

vlw cara