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In various questions regarding why $\frac{dy}{dx}$ can't be treated as a ratio, the main arguement used in most of the answer is: "And because we cannot express this limit-of-a-quotient as a-quotient-of-the-limits".

Why the limit of a quotient is not the quotient of a limit ?

  • A better question: Why would it be? Alternatively, write down a bunch of limits of quotients and see they aren't quotients of the corresponding limits. –  Jan 27 '16 at 03:52
  • @T.Bongers: I am asking why it is (for the intuive feel for it). –  Jan 27 '16 at 04:03
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    I always find it strange when people ask for an intuitive reason for why something is false. I think a vastly better place to start is addressing any (necessarily incorrect) intuition that the statement is true. –  Jan 27 '16 at 04:06
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    The limit of a quotient is the quotient of the limits, if the downstairs limit is nonzero. And if the limit of the denominator is indeed zero, the quotient of the limits is not defined. – Lubin Jan 27 '16 at 04:11
  • As per @lubin, see limit rules at:http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx and with regards to $dy/dx$ ratio, see: http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio – NoChance Jan 27 '16 at 04:15

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The limits of the numerator and denominator might not both exist, or both exist but the quotient of the limits might be undefined.

Examples:

  • $\lim_{n\to\infty} \dfrac n {n+1}$
  • $\lim_{h\to 0} \dfrac {(x+h)-x} h$: both limits exist, but $0/0$ is undefined.
BrianO
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