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Math Help - Limits at Infinity

  1. #1
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    Limits at Infinity

    Evaluate the limit at infinity. f(x) = 4[(x + sin x)/(x)]

    lim x→∞ f(x)

    The limit of sinx as x approaches infinity = does not exist, which leaves me with x/x or ∞/∞ = 1 so 4(1)=4?

    Thanks in advance for the help!
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  2. #2
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    Re: Limits at Infinity

    Quote Originally Posted by Johnny Walker Black View Post
    Evaluate the limit at infinity. f(x) = 4[(x + sin x)/(x)]

    lim x→∞ f(x)

    The limit of sinx as x approaches infinity = does not exist, which leaves me with x/x or ∞/∞ = 1 so 4(1)=4?

    Thanks in advance for the help!
    For large x:
    x >> |\sin(x)|

    Therefore your sin(x) term is negligible


    Edit: Visualise by thinking of a suitably large number - say 10^6 then \sin(x) \approx -0.35.

    Now what is 10^6 - (-0.35)
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  3. #3
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    Re: Limits at Infinity

    Quote Originally Posted by Johnny Walker Black View Post
    Evaluate the limit at infinity. f(x) = 4[(x + sin x)/(x)]

    lim x→∞ f(x)

    The limit of sinx as x approaches infinity = does not exist, which leaves me with x/x or ∞/∞ = 1 so 4(1)=4?

    Thanks in advance for the help!
    f(x)= 4\frac{x+ sin(x)}{x}= 4(1+ \frac{sin(x)}{x})
    Now, -1\le sin(x)\le 1 for all x so as the denominator goes to infinity, \frac{sin(x)}{x} goes to 0.
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