Question

resolva:

$\lim _{x\to \:1}\left(\frac{\sqrt{x}-1}{x-1}\right)$
​

Answer 1

$\displaystyle \lim_{\text x \to 1} \frac{\sqrt{\text x}-1}{\text x-1}$

Se substituirmos x =1, vai dar indeterminação.. então a ideia vai ser manipular essa expressão de alguma forma.

Podemos racionalizar o numerador da seguinte forma :

$\displaystyle \lim_{\text x \to 1} \frac{(\sqrt{\text x}-1)}{(\text x-1)}.\frac{(\sqrt{\text x}+1)}{(\sqrt{\text x}+1)}$

sendo  $(\sqrt{\text x}-1).(\sqrt{\text x}+1) = \text x - 1$, então :

$\displaystyle \lim_{\text x \to 1} \frac{(x-1)}{(x-1).(\sqrt{\text x}+1) }$

$\displaystyle \lim_{\text x \to 1} \frac{1}{\sqrt{\text x}+1 }$

substituindo x = 1 :

$\displaystyle \displaystyle \lim_{\text x \to 1} \frac{1}{\sqrt{\text x}+1 } \to \frac{1}{\sqrt{\text 1}+1 } = \frac{1}{2}$

Portanto :

$\huge\boxed{\displaystyle \lim_{\text x \to 1} \frac{\sqrt{\text x}-1}{\text x-1} = \frac{1}{2}}$

OU

Ao substituir x =1 dá indeterminação do tipo $\displaystyle \frac{0}{0}$ , então podemos derivar o numerador e denominador até sumir a indeterminação, ou seja :

$\displaystyle \lim_{\text x \to 1} \frac{[\sqrt{\text x}-1]'}{[\text x-1]'}$

sabendo que :

$\displaystyle [\sqrt{\text x}]' = [\text x^{\displaystyle \frac{1}{2}}]' = \frac{1}{2}.\frac{1}{\sqrt{x}}$

então :

$\displaystyle \lim_{\text x \to 1} \frac{[\sqrt{\text x}-1]'}{[\text x-1]'} = \lim_{\text x \to 1}\frac{\displaystyle \frac{1}{2\sqrt{x}}}{1} = \lim_{\text x \to 1} \frac{1}{2\sqrt{x}}$

substituindo x = 1 :

$\displaystyle \lim_{\text x \to 1} \frac{1}{2\sqrt{\text x}}= \frac{1}{2\sqrt{1}}=\frac{1}{2}$

Answer 2

$\sf lim_{x \to1}\Bigg( \frac{ \sqrt{x} - 1 }{x - 1} \Bigg) \\ \\ \\ \\ \\ \sf lim_{x \to1}\Big( \sqrt{x} - 1\Big) \\ \sf lim_{x \to1}\Big(x - 1\Big) \\ \\ \\ \\ \sf 0 \\ \sf 0 \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \sqrt{x} - 1}{x - 1} \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{d}{dx}\Big( \sqrt{x} - 1\Big) }{ \frac{d}{dx}\Big(x - 1\Big) } \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{d}{dx}\Big( \sqrt{x} \Big) - \frac{d}{dx} (1)}{ \frac{d}{dx}(x - 1) } \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{1}{2 \sqrt{x} } - \frac{d}{dx}(1) }{ \frac{d}{dx}(x - 1) } \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{1}{2 \sqrt{x} } - 0}{ \frac{d}{dx} (x - 1)} \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{1}{2 \sqrt{x} } }{ \frac{d}{dx} (x - 1)} \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{1}{2 \sqrt{x} } }{ \frac{d}{dx} (x) - \frac{d}{dx }(1) } \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{1}{2 \sqrt{x} } }{1 - \frac{d}{dx}(1) } \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{1}{2 \sqrt{x} } }{1 - 0} \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{ \frac{1}{2 \sqrt{x} } }{1} \Bigg) \\ \\ \\ \\ \sf lim_{x \to1}\Bigg( \frac{1}{2 \sqrt{x} } \Bigg)\\ \\ \\ \\ \sf \frac{ lim_{x \to1}(1) }{ lim_{x \to1} \Big(2 \sqrt{x} \Big)} \\ \\ \\ \\ \sf \frac{1}{ lim_{ x \to1 }\Big(2 \sqrt{x} \Big)} \\ \\ \\ \\ \sf \frac{1}{2 \cdot lim_{x \to1}\Big( \sqrt{x} \Big)} \\ \\ \\ \\ \sf \frac{1}{ \sqrt[2]{ lim_{x \to1}(x) } } \\ \\ \\ \sf \frac{1}{2 \sqrt{1} } \\ \\ \\ \\ \sf \frac{1}{2 \cdot1} \\ \\ \\ \\ \sf {\color{red}\frac{1}{2} }$

Att: Nerd1990