[tex]\displaystyle\lim_{x\to-2}\frac{\sqrt{3x+10}-2}{x+2}[/tex]
Rationalize the numerator:
[tex]\dfrac{\sqrt{3x+10}-2}{x+2}\cdot\dfrac{\sqrt{3x+10}+2}{\sqrt{3x+10}+2}=\dfrac{(3x+10)-4}{(x+2)(\sqrt{3x+10}+2)}=\dfrac3{\sqrt{3x+10}+2}[/tex]
where the [tex]x+2[/tex] factors in the numerator and denominator cancel. Then the limit is
[tex]\displaystyle\lim_{x\to-2}\frac{\sqrt{3x+10}-2}{x+2}=\lim_{x\to-2}\frac3{\sqrt{3x+10}+2}=\frac3{\sqrt4+2}=\boxed{\frac34}[/tex]
