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Math Help - limits

  1. #1
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    limits

    how do you solve lim ( sin x/ x) as

    (a) x tends to 1
    (b)x tends to infinity?



    without using l'hosipital rule..



    can i say that for part (a) since the lim of top and bottom of the fraction exiist, lim ( sinx/ x) = lim ( sinx ) lim (1/x)? then im stuck alr...
    thanks
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    how do you solve lim ( sin x/ x) as

    (a) x tends to 1
    (b)x tends to infinity?



    without using l'hosipital rule..



    can i say that for part (a) since the lim of top and bottom of the fraction exiist, lim ( sinx/ x) = lim ( sinx ) lim (1/x)? then im stuck alr...
    thanks

    Yes, you can say that because the limits exists AND ARE FINITE, so \lim_{x\to 1}\frac{\sin x}{x}=\sin 1

    Now for (b) prove the following easy lemma:

    If f(x)\xrightarrow [x\to x_0] {} 0\,\,and \,\,g(x) bounded in some neighborhood of  x_0 , then \lim_{x\to x_0}f(x)g(x)=0 , and this lemma includes the case x_0=\infty

    Tonio

    Pss. BTW, the condition not to use L'H rule is pointless: you can NOT use L'H in none of both cases.
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  3. #3
    Super Member Deadstar's Avatar
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    Could you use the Sandwich Theorem for part b?

    \frac{-1}{x} \leq \frac{\sin(x)}{x} \leq \frac{1}{x}.

    Letting x \rightarrow \infty gives you...

    0 \leq \frac{\sin(x)}{x} \leq 0 so \lim_{x \rightarrow \infty} \frac{\sin(x)}{x} = 0..?

    (This is sort of directed to Tonio by the way but is another method to look at)
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by tonio View Post
    you can NOT use L'H in none of both cases.
    Nice.
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  5. #5
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    Quote Originally Posted by Deadstar View Post
    Could you use the Sandwich Theorem for part b?

    \frac{-1}{x} \leq \frac{\sin(x)}{x} \leq \frac{1}{x}.

    Letting x \rightarrow \infty gives you...

    0 \leq \frac{\sin(x)}{x} \leq 0 so \lim_{x \rightarrow \infty} \frac{\sin(x)}{x} = 0..?

    (This is sort of directed to Tonio by the way but is another method to look at)

    Cool!...and correct, of course.

    Tonio
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