Saturday, October 17, 2009

Importance of Definitions (Zero)

Evaluate the following and justify your solution. Are there questions that are contradictory, paradoxical, and/or nonsensical?

If the equations in latex do not display properly, enable javascript in your browser.
  1. \(0^0 = \)
  2. \( \lim \limits_{x \to 0} x^0 = \)
  3. \( \lim \limits_{x\to 0} 0^x = \)
  4. \( \lim \limits_{x\to 0} x^x = \)
  5. \( \frac{0^a}{0^a} = \)
  6. \( 0^{a-a} = \)
  7. \( \frac{a^0}{a^0} = \)
  8. \( a^{0-0} = \)
  9. \( \lim \limits_{x\to 0} \frac{x^a}{x^a} = \)
  10. \( 0^\infty = \)
  11. \( \lim \limits_{x\to 0} 0^{\frac{1}{x}} = \)
  12. \( \lim \limits_{x\to 0} \sqrt[x]{0} = \)
  13. \( \lim \limits_{x\to 0} x^{\frac{1}{x}} = \)
  14. \( \lim \limits_{x\to 0} \sqrt[x]{x} = \)
  15. If \( 0^0=a \), what is \( log_{0}a = \)
  16. If \( 0^0=a \), what is \( \lim \limits_{x \to 0} log_{x}a = \)
  17. If \( 0^0=a \), what is \( \sqrt[0]{a} = \)
  18. If \( 0^0=a \), what is \( \lim \limits_{x \to 0} \sqrt[x]{a} = \)
UPDATE: Reposted here. Replaced \mathop with \lim \limits_ in latex code.

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