Question

Qual é o lim de X tendendo a 7 de:
Raiz quadrada de X +2, menos 3.Dividido por x -7?

Answer 1

$\large\begin{array}{lr}\displaystyle\rm\lim_{x\to7} \, \frac{\sqrt{x+2}-3}{x-7} = \frac{0}{0}~~\boxtimes\\\\ \begin{aligned} \displaystyle\rm\lim_{x\to7}\, \frac{\sqrt{x+2}-3}{x-7} &= \displaystyle\rm\lim_{x\to7}\, \frac{\sqrt{x+2}-3}{x-7} \cdot \dfrac{\sqrt{x+2}+3}{\sqrt{x+2}+3} \\\\&=\displaystyle\rm \lim_{x\to7} \,\frac{x+2-9}{x-7\cdot\sqrt{x+2}+3} \\\\&=\displaystyle\rm \lim_{x\to7}\, \frac{\cancel{x-7}}{\cancel{x-7}\cdot\sqrt{x+2}+3} \\\\&=\displaystyle\rm \lim_{x\to7}\, \frac{1}{\sqrt{x+2}+3} \\\\&=\displaystyle\rm \frac{1}{\sqrt{7+2}+3} \\\\&=\displaystyle\rm \frac{1}{\sqrt{9} + 3} \end{aligned} \\\\\red{\underline{\boxed{\boxed{\rm \therefore\: \displaystyle\rm\lim_{x\to7} \,\frac{\sqrt{x+2}-3}{x-7} = \frac{1}{6} }}}} \\\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\blacksquare \end{array}$