Question

Lim com x tendendo a 2 de 10^(x-2)-1/x-2

Answer 1

Temos o seguinte:

$\lim_{x \to 2} \dfrac{10^{x-2} - 1}{x-2}$

Aplicando L'Hopital:

$\lim_{x \to 2} \dfrac{10^{x-2} - 1}{x-2} = \lim_{x \to 2} \dfrac{[10^{x-2} - 1]'}{[x-2]'}$

Usando:

$\dfrac{d}{dx} A^x = ln(A) \cdot A^x$

Logo:

$\lim_{x \to 2} \dfrac{[10^{x-2} - 1]'}{[x-2]'} = \lim_{x \to 2} \dfrac{ln(10) \cdot 10^{x-2}}{1} = \lim_{x \to 2} ln(10) \cdot 10^{x-2} \\ \\ \\ \lim_{x \to 2} ln(10) \cdot 10^{x-2} = \lim_{x \to 2} ln(10) \cdot 10^{2-2} = ln(10) \cdot 1 = \boxed{ln(10)}$