Question

urgente
Prove cada proposição utilizando a definição , de limite.​

Answer 1

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A) Verdadeiro.

$\bold{\underset{x\rightarrow2}{\lim}~}( {x}^{2} - 4x + 5) = 1$

$\bold{\underset{x\rightarrow2}{\lim}~}( {x}^{2} - 4x) + \bold{\underset{x\rightarrow2}{\lim}~}(5)$

$\bold{\underset{x\rightarrow2}{\lim}~}( {x}^{2}) - \bold{\underset{x\rightarrow2}{\lim}~}(4x) + \bold{\underset{x\rightarrow2}{\lim}~}(5)$

$\bold{\underset{x\rightarrow2}{\lim}~}( {x}^{2} ) - \bold{\underset{x\rightarrow2}{\lim}~}(4x) + 5$

$(\bold{\underset{x\rightarrow2}{\lim}~}(x) {)}^{2} - \bold{\underset{x\rightarrow2}{\lim}~}(4x) + 5$

$(\bold{\underset{x\rightarrow2}{\lim}~}(x) {)}^{2} - 4 \times \bold{\underset{x\rightarrow2}{\lim}~}(x) + 5$

${2}^{2} - 4 \times \bold{\underset{x\rightarrow2}{\lim}~}(x) + 5$

${2}^{2} - 4 \times 2 + 5$

$4 - 4 \times 2 + 5$

$4 - 8 + 5$

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B) Verdadeiro.

$\bold{\underset{x\rightarrow - 2}{\lim}~}( {x}^{2} - 1)$

$\bold{\underset{x\rightarrow - 2}{\lim}~}( {x}^{2} ) - \bold{\underset{x\rightarrow - 2}{\lim}~}(1)$

$(\bold{\underset{x\rightarrow - 2}{\lim}~}(x) {)}^{2} - \bold{\underset{x\rightarrow - 2}{\lim}~}(1)$

$(\bold{\underset{x\rightarrow - 2}{\lim}~}(x) {)}^{2} - 1$

$( - 2 {)}^{2} - 1$

$4 - 1$

$\boxed{ \color{green} \boxed{{ 3 }}}$

C) Verdadeiro.

$\bold{\underset{x\rightarrow3}{\lim}~}( \frac{ {x}^{2} + x - 12}{x - 3}$

$\bold{\underset{x\rightarrow3}{\lim}~}( {x}^{2} + x - 12) \\ \bold{\underset{x\rightarrow3}{\lim}~}(x - 3) \: \: \: \: \: \: \: \: \: \:$

${3}^{2} + 3 - 12 \\ \bold{\underset{x\rightarrow3}{\lim}~}(x - 3)$

$0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \bold{\underset{x\rightarrow3}{\lim}~}(x - 3)$

$0 \: \: \: \: \: \: \\ 3 - 3$

$0 \\ 0$

$\bold{\underset{x\rightarrow3}{\lim}~}( \frac{ {x}^{2} + x - 12}{x - 3} )$

$\bold{\underset{x\rightarrow3}{\lim}~}( \frac{ {x}^{2} + 4x - 3x - 12}{x - 3} )$

$\bold{\underset{x\rightarrow3}{\lim}~}( \frac{x \times (x + 4) - 3x - 12}{x - 3} )$

$\bold{\underset{x\rightarrow3}{\lim}~}( \frac{x \times (x + 4) - 3(x + 4)}{x - 3} )$

$\bold{\underset{x\rightarrow3}{\lim}~}( \frac{(x + 4) \times (x - 3)}{x - 3} )$

$\bold{\underset{x\rightarrow3}{\lim}~}( \frac{(x + 4) \times \cancel{ (x - 3) }}{\cancel{ x - 3 }})$

$\bold{\underset{x\rightarrow3}{\lim}~}(x + 4)$

$\bold{\underset{x\rightarrow3}{\lim}~}(x) + \bold{\underset{x\rightarrow3}{\lim}~}(4)$

$3 + \bold{\underset{x\rightarrow3}{\lim}~}(4)$

$3 + 4$

$\boxed{ \color{green} \boxed{{ 7 }}}$

D) Verdadeiro.

$\bold{\underset{x\rightarrow9}{\lim}~}( \sqrt{9 - x} )$

$\sqrt{\bold{\underset{x\rightarrow9}{\lim}~}(9 - x)}$

$\sqrt{\bold{\underset{x\rightarrow9}{\lim}~}(9) - \bold{\underset{x\rightarrow9}{\lim}~}(x)}$

$\sqrt{9 - \bold{\underset{x\rightarrow9}{\lim}~}(x)}$

$\sqrt{9 - 9}$

$\sqrt{0}$

$\boxed{ \color{green} \boxed{{ 0 }}}$

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${\boxed{ \color{blue} \boxed{ 26 |02|22  }}}{\boxed{ \color{blue} \boxed{Espero \:  ter  \: ajudado \: ☆}}}$