Question

Para cada uma das seguintes funções, ache: (figura)

Answer 1

$a)\\ \lim_{x \to 2} \frac{3x^2-12}{x-2} \\\\Fatorando \;3x^2-12\;em\;3(x-2)(x+2):\\\\\lim_{x \to 2} \frac{3(x-2)(x+2)}{x-2} \\\\Eliminamos\;a\;indetermina\c{c}\~{a}o\;do\;denominador, cortando\;os\;(x-2):\\\\\lim_{x \to 2}\;3(x+2)\\\lim_{x \to 2}\;3x+6\\\lim_{x \to 2}\;6 +6 = \boxed{12}$

$b)\\\\ \lim_{x \to 2} \frac{\frac{1}{x}-\frac{1}{2} }{x-2}\\\\Aplicando\;as\;propriedades\;das\;fra\c{c}\~{o}es:\\\frac{1}{x}-\frac{1}{2} = \frac{2-x}{2x}\\\\Temos\;que:\\\\\lim_{x \to 2} \frac{\frac{2-x}{2x}}{x-2}\\\\\lim_{x \to 2} \frac{2-x}{2x(x-2)}}\\\\Temos\;no\;numerador\;um\;(2-x)\;e\;no\;denominador\;(x-2). Podemos\;cortar\\\;esses\;dois\;termos,pois\;qualquer\;que\;seja\;o\;x,o\;resultado\;desse\;quociente\\\;sera\;-1:\\\lim_{x \to 2} -\frac{1}{2x}}\\\\Substituindo\;os\;valores:$

$\lim_{x \to 2}-\frac{1}{2x}}=\boxed{-\frac{1}{4}}$

$c) \lim_{x \to 2}\;\frac{(3x^2+5x-1)-(12+10-1)}{x-2}}\\\lim_{x \to 2}\;\frac{3x^2+5x-1-21}{x-2}}\\\lim_{x \to 2}\;\frac{3x^2+5x-22}{x-2}}\\\\Fatorando\;3x^2+5x-22\;temos\;(x-2)(3x+11).Para\;chegar\;a\;esse\;resultado\;\\eu\;usei\;o\;metodo\;Briot-Ruffini,\;dividindo\;por\;(x-2)\\\\\lim_{x \to 2}\;\frac{(x-2)(3x+11)}{x-2}}\\\lim_{x \to 2}\;3x+11\\\lim_{x \to 2}\;6+11=\boxed{17}$

$d)\\\lim_{x \to 2}\;\frac{\frac{1}{x+1} - \frac{1}{3}}{x-2}}\\\\\frac{1}{x+1} - \frac{1}{3} = \frac{2-x}{3x+3} \\\\\lim_{x \to 2}\;\frac{\frac{2-x}{3x+3}}{x-2}}\\\\\lim_{x \to 2}\;\frac{2-x}{3x+3(x-2)}\\\\\lim_{x \to 2}\;-\frac{1}{3x+3}\\\lim_{x \to 2}\;-\frac{1}{6+3} = \boxed{-\frac{1}{9}}$