Prove by definition (epsilon-delta) the next limits:
1. lim(x-->inf) {(2x^2-11x-11)/(4x^2+11x-1)} = 1/2
2. lim(x-->1) {(x^3-3x)/(x^2-2x+1)} = -inf
Thanks!
For the first one note that, . Now, . So, let be given. Choosing ensures that . We may conclude that . Using similar techniques we may conclude that . We may therefore conclude that
For the second one, note that which for a sufficiently small neighborhood around is strictly less than . So, let be given, then choosing ensures that . The conclusion follows.
P.S. Technically, the second should be