What is $lim_(xrarr-3) (2x+6)/|x+3|$ ?

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Answer 1 · 2016-07-07T14:03:20

$lim_(xrarr-3^-) (2x+6)/|x+3| = -2$

$lim_(xrarr-3^+) (2x+6)/|x+3| = +2$

$lim_(xrarr-3) (2x+6)/|x+3|$

let $x = -3 + h, 0 < abs h "<<" 1$

so the lim becomes

$lim_( h to 0) (2(-3+h)+6)/|(-3+h)+3|$

$=lim_( h to 0) (2h)/(|h|) =lim_( h to 0) 2 h/abs h$

for the left sided limit, h < 0 so we have $"-ve"/"+ve" = "-ve"$

so $lim_(xrarr-3^-) (2x+6)/|x+3| = -2$

for the right sided limit, h > 0 so we have $"+ve"/"+ve" = "+ve"$

so $lim_(xrarr-3^+) (2x+6)/|x+3| = +2$

What is lim_(xrarr-3) (2x+6)/|x+3|lim_(xrarr-3) (2x+6)/|x+3|?