Question

Lim x—> -2
sqrt(3x+10) -2/ x+2

Answer 1

`\displaystyle\lim_(x\to-2)(√(3x+10)-2)/(x+2)`

Rationalize the numerator:

`(√(3x+10)-2)/(x+2)\cdot(√(3x+10)+2)/(√(3x+10)+2)=((3x+10)-4)/((x+2)(√(3x+10)+2))=\frac3{√(3x+10)+2}`

where the
`x+2` factors in the numerator and denominator cancel. Then the limit is

`\displaystyle\lim_(x\to-2)(√(3x+10)-2)/(x+2)=\lim_(x\to-2)\frac3{√(3x+10)+2}=\frac3{\sqrt4+2}=\boxed{\frac34}`