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If the limit of a derivative of a function when x tending to infinity exists, it is zero. ( Considering limit of the function at x tending to infinity is finite.)

I wonder what this limit implies, physically or graphically.

aarbee
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    The assertion is not true as written, the limit of the derivative need not exist. If it exists, then it is $0$. – Daniel Fischer Nov 02 '16 at 15:05
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    I think one intuitive meaning could be that the function's graph "flattens down" towards the horizontal line $;y=L;$ ...and this line's derivative is zero, indeed. – DonAntonio Nov 02 '16 at 15:05
  • @DonAntonio- Actually this is what my doubt was. Are we saying that the limit of derivative when x tending to infinity represents taking derivative once again? I wonder how could limit imply that. – aarbee Nov 02 '16 at 15:53
  • @Daniel, thanks. I have edited my question. Hope it is correct now. – aarbee Nov 02 '16 at 15:57
  • Not quite, the existence of the (finite) limit of the function itself is important. Otherwise look at $f(x) = x$, the limit of the derivative is $1$. – Daniel Fischer Nov 02 '16 at 15:59
  • @Daniel, thanks once again. Edited accordingly. – aarbee Nov 02 '16 at 16:35

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As Daniel pointed out, your assertion is incorrect. If a function approaches some value L (presumably a horizontal asymptote) when $x$ goes to infinity, then the derivative does not have to go to zero at all. Under "high school" circumstances, it usually will (most of those functions behave kind of "nice"?), however, consider the following counter example: $y=\frac{sinx^2}{x}$. It is clear that for large $x$, the $y$ goes to zero. The derivative of this function is $y'=\frac{2x^2cosx^2-sinx^2}{x^2}$ which simplifies to $2cosx^2-\frac{sinx^2}{x^2}$. Is it clear that the derivative will never go to zero for larger $x$?

imranfat
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