limit

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DATE: Sept. 14, 2012, 4:13 a.m.

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  1. protip: http://www.wolframalpha.com/input/?i=lim+h-%3E0+%28%282%28x%2Bh%29%2F%28x%2Bh%2B1%29%5E2%29+-+%282x%2F%28x%2B1%29%5E2%29%29%2Fh
  3. lim_(h->0) ((2 (h+x))/(h+x+1)^2-(2 x)/(x+1)^2)/h
  4. Factor the numerator and denominator:
  5. = lim_(h->0) (h (-2 (h x+x^2-1)))/((((x+1)^2 (h+x+1)^2) (h ((x+1)^2 (h+x+1)^2)))/((x+1)^2 (h+x+1)^2))
  6. Cancel terms, assuming h/((x+1)^2 (h+x+1)^2)=!=0:
  7. = lim_(h->0) -(2 (h x+x^2-1))/((x+1)^2 (h+x+1)^2)
  8. Factor out constants:
  9. = -(2 (lim_(h->0) (x^2+h x-1)/(h+x+1)^2))/(x+1)^2
  10. The limit of (h x+x^2-1)/(h+x+1)^2 as h approaches 0 is (x-1)/(x+1):
  11. = -(2 (x-1))/(x+1)^3
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