Question

$\lim_{x \to 0} \frac{12^{x} -3 ^{x} }{3x}$

Alguém poderia me ajudar passo-a-passo?
A resposta é [ ( ln4 ) / ( 3 ) ].

Answer 1

Resposta:  $\mathsf{\dfrac{1}{3}\,\ell n(4).}$

Explicação passo-a-passo:

Calcular o limite

    $\mathsf{\underset{x\to 0}{\ell im}~\dfrac{12^x-3^x}{3x}}\\\\\\ \mathsf{=\underset{x\to 0}{\ell im}~\dfrac{1}{3}\cdot \dfrac{12^x-3^x}{x}}\\\\\\ \mathsf{=\dfrac{1}{3}\cdot \underset{x\to 0}{\ell im}~\dfrac{12^x-3^x}{x}}$

Coloque 3^x em evidência no numerador:

    $\mathsf{=\dfrac{1}{3}\cdot \underset{x\to 0}{\ell im}~\dfrac{3^x\cdot (\frac{12^x}{3^x}-1)}{x}}\\\\\\ \mathsf{=\dfrac{1}{3}\cdot \underset{x\to 0}{\ell im}~\dfrac{3^x\cdot ((\frac{12}{3})^x-1)}{x}}\\\\\\ \mathsf{=\dfrac{1}{3}\cdot \underset{x\to 0}{\ell im}~3^x\cdot \dfrac{4^x-1}{x}}\\\\\\ \mathsf{=\dfrac{1}{3}\cdot 3^0\cdot \ell n(4)}\\\\\\ \mathsf{=\dfrac{1}{3}\,\ell n(4)\quad\longleftarrow\quad resposta.}$

Dúvidas? Comente.

Bons estudos! :-)