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  1. #1
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    help with limits

    How do I evaluate this limit?
    lim x*sin(1/x)
    x->infinity
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  2. #2
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    Re: help with limits

    Rewrite:
    \lim_{x\to \infty} \sin\left(\frac{1}{x}\right)\cdot x=\lim_{x\to \infty} \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}}

    You can use the substitution, let \frac{1}{x}=u. If x\to \infty then u \to ?, afterwards you should recognize a standard limit.
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    Re: help with limits

    Quote Originally Posted by Sneaky View Post
    How do I evaluate this limit?
    lim x*sin(1/x)
    x->infinity
    Just note that x\sin \left( {\frac{1}{x}} \right) = \frac{{\sin \left( {\frac{1}{x}} \right)}}{{\frac{1}{x}}}.

    \lim _{h \to 0} \frac{{\sin (h)}}{h} = ?
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    Re: help with limits

    sin(h) / h as x-> 0 should be 0, but how do I formally show that? And why are we considering x->0 when we need x->infinity?

    edit: after its sin(h)/h, can I use L'hopital rule and derive the top and bottom to get 1*cos(h) / 1, then then sub h as 1/x, then sub x as infinity, which results to 0, and cos of 0 is 1, so the limit is 1?
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    Re: help with limits

    Quote Originally Posted by Sneaky View Post
    sin(h) / h as x-> 0 should be 0, but how do I formally show that? And why are we considering x->0 when we need x->infinity?

    edit: after its sin(h)/h, can I use L'hopital rule and derive the top and bottom to get 1*cos(h) / 1, then then sub h as 1/x, then sub x as infinity, which results to 0, and cos of 0 is 1, so the limit is 1?
    Do not use L'hopital rule on these problems.

    \lim _{h \to 0} \frac{{\sin (h)}}{h} is the same as
    \lim _{x \to \infty } x\sin \left( {\frac{1}{x}} \right) if h = \frac{1}{x}.
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    Re: help with limits

    Quote Originally Posted by Sneaky View Post
    sin(h) / h as x-> 0 should be 0, but how do I formally show that? And why are we considering x->0 when we need x->infinity?

    edit: after its sin(h)/h, can I use L'hopital rule and derive the top and bottom to get 1*cos(h) / 1, then then sub h as 1/x, then sub x as infinity, which results to 0, and cos of 0 is 1, so the limit is 1?
    sin(h) / h as h-> 0 is a standard limit. You are expected to review your class notes and textbook as part of the process in solving questions like the one you posted (and certainly after getting help with the question).
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    Re: help with limits

    Okay, I now understand why we get
    lim h-> 0 sin(h)/h, but I'm still not sure why the limit of this is 0, if you plug in 0 in h, then you get 0/0.
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    Re: help with limits

    Quote Originally Posted by Sneaky View Post
    Okay, I now understand why we get
    lim h-> 0 sin(h)/h, but I'm still not sure why the limit of this is 0, if you plug in 0 in h, then you get 0/0.
    The limit is NOT 0. As I have already said, go to your class notes or textbook - review the standard limits you have undoubtedly been given and are expected to know and use.
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  9. #9
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    Re: help with limits

    Or better yet, do a little research.

    Think of the unit circle.



    If the radius is one unit in length and angle made from the radius and the horizontal axis (measured in radians) is \displaystyle \theta, then the green length is \displaystyle \sin{\theta}, the red length is \displaystyle \cos{\theta}, and the purple length is \displaystyle \tan{\theta}.

    Clearly, the area of the smallest triangle is a little less than the area of the circular sector, which is a little less than the area of the larger triangle. So

    \displaystyle \begin{align*} \frac{1}{2}\sin{\theta}\cos{\theta} \leq \frac{1}{2}\theta &\leq \frac{1}{2}\tan{\theta} \\ \sin{\theta}\cos{\theta} \leq \theta &\leq \tan{\theta} \\ \sin{\theta}\cos{\theta} \leq \theta &\leq \frac{\sin{\theta}}{\cos{\theta}} \\ \cos{\theta} \leq \frac{\theta}{\sin{\theta}} &\leq \frac{1}{\cos{\theta}} \\ \frac{1}{\cos{\theta}} \geq \frac{\sin{\theta}}{\theta} &\geq \cos{\theta} \\ \cos{\theta} \leq \frac{\sin{\theta}}{\theta} &\leq \frac{1}{\cos{\theta}} \\ \lim_{\theta \to 0}\cos{\theta} \leq \lim_{\theta \to 0}\frac{\sin{\theta}}{\theta} &\leq \lim_{\theta \to 0}\frac{1}{\cos{\theta}} \\ 1 \leq \lim_{\theta \to 0}\frac{\sin{\theta}}{\theta} &\leq 1 \end{align*}

    It should be clear therefore, that \displaystyle \lim_{\theta \to 0}\frac{\sin{\theta}}{\theta} = 1.

    (Actually, we have really only proven the right-hand limit, but the proof of the left-hand limit is almost identical.)
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