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If $f$ is continuous in $[0,1]$ then $$\displaystyle \lim_ {n \to \infty}   \sum_{j=0}^{[n/2]} \frac{1}{n} f \left(\frac{j}{n} \right)$$ (where $[y]$ is the largest integer less than or equal to $y$)

  1. does not exist
  2. exists and is equal to $\frac{1}{2} \int_0^1 f(x) dx$
  3. exists and is equal to $ \int_0^1 f(x) dx$
  4. exists and is equal to $\int_0^{1/2} f(x) dx$
in Calculus
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