(a) Factorise: (2x+5)(x+1)/((x+3)(x+4). No common factors between numerator and denominator.
As x→-3, denominator→0, but numerator evaluates to 2, so limit is undefined (infinity).
(b) Factorise: (x-2)(x+2)/(x(x-2)). When x=-2, numerator=0, denominator≠0. No anomalies, so limit is zero.
(c) Factorise: (x-3)(2x+1)/((x-3)(x+3)). Common factor x-3 shared between numerator and denominator. The expression becomes (2x+1)/(x+3) except for x=3. But as x→3, the expression→7/6. Therefore the limit is 7/6.
(d) The denominator can be written (125-x3)/(125x3) making the whole expression, after factoring:
125x3(x-5)/((5-x)(25+5x+x2). x-5 is a common factor reducing the expression when x≠5:
-125x3/(25+5x+x2). When x→5, limit→-625/3.