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Practice Problems for 4/10

  1. A citrus farmer usually plants 30 orange trees per acre, and each tree yields 500 oranges when harvested. For each additional tree per acre, each tree produces 10 fewer oranges. How many trees should be planted per acre to maximize the harvest, and what is the yield of this harvest?

  2. Which is correct, and WHY?
    1. \( \lim_{x\rightarrow 5}\frac{x-5}{x^3-5}=\lim_{x\rightarrow 5} \frac{1}{3x^2}=\frac{1}{75}\)
    2. \( \lim_{x\rightarrow 5}\frac{x-5}{x^3-5}=\frac{0}{120}=0\)

  3. For each, calculate the limit, or state why it does not exist.
    1. \(\lim_{x\rightarrow \infty} e^{-x}\ln x\)
    2. \(\lim_{x\rightarrow \infty} x-\sqrt{x^2-1}\)
    3. \(\lim_{x\rightarrow 0} \frac{e^x-1}{x^2}\)