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Thread: Limit Involves Sine as x goes to Infinite

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    Limit Involves Sine as x goes to Infinite

    Find the limit of the following functions as x goes to Infinity.

    (a) f(x) = sin[ (x^2 + 4) / (x^2 - 4) ]

    (b) f(x) = (4 sin x)/x
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    Re: Limit Involves Sine as x goes to Infinite

    Quote Originally Posted by joshuaa View Post
    Find the limit of the following functions as x goes to Infinity.

    (a) f(x) = sin[ (x^2 + 4) / (x^2 - 4) ]

    (b) f(x) = (4 sin x)/x
    for (a), the limit is $sin(1)$

    for (b), the limit is 0
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    Re: Limit Involves Sine as x goes to Infinite

    wOW. That was surprising. Even for the angle, I can apply the rule of thumb.
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    Horizontal and Vertical Asymptotes (2)

    Does this function have any horizontal and vertical asymptotes?

    f(x) = sin[ (x^2 + 4) / (x^2 - 4) ]
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    Re: Horizontal and Vertical Asymptotes (2)

    Quote Originally Posted by joshuaa View Post
    Does this function have any horizontal and vertical asymptotes?

    f(x) = sin[ (x^2 + 4) / (x^2 - 4) ]
    When looking for vertical asymptotes the first thing to look at is the denominator. If the denominator goes to 0 then you might have a vertical asymptote. Further, if the numerator does not go to 0 at the same points as the denominator, then you have a vertical asymptote. There are two of these here. Can you find them?

    As for horizontal asymptotes you want to take the limit as x goes to either positive or negative infinity. As x gets large what happens to the numerator? To the denominator?

    -Dan
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    Re: Limit Involves Sine as x goes to Infinite

    x = -2 and x = 2 make the denominator equal to zero. When I look at the graph I see a very bold oscillation at these two points, but I don't see clearly the function increasing or decreasing without bound at them.
    Can these two points be vertical asymptotes? What are their direction, positive infinity or negative infinity?


    When I take the limit as x goes to infinity or negative infinity, I get the same answer which is sin(1).

    sin(1) = y = horizontal asymptotes
    Last edited by joshuaa; Oct 13th 2016 at 08:06 AM.
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    Re: Limit Involves Sine as x goes to Infinite

    no vertical asymptotes ... as $x \to \pm 2$ the argument of the sine function gets very large positive or negative causing an increasing rate of oscillation between $y=-1$ and $y=1$ as $x$ gets closer to $\pm 2$
    Attached Thumbnails Attached Thumbnails Limit Involves Sine as x goes to Infinite-sine_func.png  
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    Re: Limit Involves Sine as x goes to Infinite

    Thanks. It is clear.
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