Question #60d52

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Answer 1 · 2018-01-09T10:50:38

Diverges

$lim_(xrarr3)(1/(x^2-9)-1/(x^3-27))$ $=$

$lim_(xrarr3)(1/(x^2-3^2)-1/(x^3-3^3))$ $=$

$lim_(xrarr3)(1/((x-3)(x+3))-1/((x-3)(x^2+3x+9)))$ $=$

$lim_(xrarr3)(x^2+3x+9-(x+3))/((x-3)(x+3)(x^2+3x+9))$ $=$

$lim_(xrarr3)(x^2+3x+9-x-3)/((x-3)(x+3)(x^2+3x+9))$ $=$

$lim_(xrarr3)(x^2+2x+6)/((x-3)(x+3)(x^2+3x+9))$

If you replace for $x=3$ you get to the $a/0$ form

$lim_(xrarr3^+)(x^2+2x+6)/((x^2-9)(x^2+3x+9))$ $=+oo$

$lim_(xrarr3^-)(x^2+2x+6)/((x^2-9)(x^2+3x+9))$ $=-oo$