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Thread: limits of slopes?

  1. #1 limits of slopes? 
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    I know there are limits like "as x approaches infinity, y approaches 2.718..."

    But could you say that a function approaches a certain slope?

    For example, , as x approaches +∞, the function approaches a slope of 1.


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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by brody View Post
    I know there are limits like "as x approaches infinity, y approaches 2.718..."

    But could you say that a function approaches a certain slope?

    For example, , as x approaches +∞, the function approaches a slope of 1.
    What you are trying to say is that the original function's first derivative f'(x) ( which is the function's slope ) approaches 1 for x -> infinity. It is still a simple limit.


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    You're trying to perfect a circle, right, as a graphed algorithm?
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    @Markus Hanke: I mean instead of saying x-->∞, f(x)-->∞ I would say for slope x-->∞, "slope"-->1. As the limits I've seen always refer to x values and their functions. Never a reference to slope. The slope of any non-linear function, I believe, is inconstant. So, you could say that, although there is no actual defined slope for the entire function, it approaches a certain, predictable steepness on the graph.

    @theQuestIsNotOver: Not at all.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by brody View Post
    @Markus Hanke: I mean instead of saying x-->∞, f(x)-->∞ I would say for slope x-->∞, "slope"-->1. As the limits I've seen always refer to x values and their functions. Never a reference to slope. The slope of any non-linear function, I believe, is inconstant. So, you could say that, although there is no actual defined slope for the entire function, it approaches a certain, predictable steepness on the graph.

    @theQuestIsNotOver: Not at all.
    You are wrong, the slope of the function is exactly defined; f'(x) = (-1/x^2)+1. It is defined for all x <> 0 ( although of course not constant ), and its limit for x -> infinity is 1; this is elemental calculus. May I ask what the point in this thread actually is ?
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    May I ask what the point in this thread actually is ?
    The point of this thread actually is for me to learn from the question I had presented as a speculative idea and you had answered it. I was never familiar with any reference to limits of slopes and I wondered if there was such a concept. As you answered, indeed there is in elementary calculus. Thank you.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by brody View Post
    May I ask what the point in this thread actually is ?
    The point of this thread actually is for me to learn from the question I had presented as a speculative idea and you had answered it. I was never familiar with any reference to limits of slopes and I wondered if there was such a concept. As you answered, indeed there is in elementary calculus. Thank you.
    No problem...remember the slope of any continuous and differentiable ( single variable ) function is just simply its first derivative, and you can use that to form limits in the usual way; so the "limit of slope" is best expressed as a limit of the first derivative.
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