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Consider the integral:

Consider the integral:. t = 2. t = 3. t = 5. ?. Example:. Example:. The function is undefined at x = 1. Since x = 1 is an asymptote, the function has no maximum. We could define this integral as:. Can we find the area under an infinitely high curve?. (left hand limit).

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Consider the integral:

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  1. Consider the integral:

  2. t = 2 t = 3 t = 5 ?

  3. Example:

  4. Example: The function is undefined at x = 1 . Since x = 1 is an asymptote, the function has no maximum. We could define this integral as: Can we find the area under an infinitely high curve? (left hand limit) We must approach the limit from inside the interval.

  5. Lets first compute the integral. Rationalize the numerator.

  6. This integral converges because the limit exists.

  7. Example: (right hand limit) We approach the limit from inside the interval. This integral diverges.

  8. Example:

  9. Example: The function approaches when .

  10. The integral converges.

  11. Example:

  12. Example

  13. Theorem

  14. Proof: (P is a constant.) What happens here? If then gets bigger and bigger as , therefore the integral diverges. If then b has a negative exponent and , therefore the integral converges.

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