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Math Help - Calculus 1: Limit of a function with sinx and cosx involved

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    Calculus 1: Limit of a function with sinx and cosx involved

    limit x---> infinity (x+sinx)/(3x+cosx)

    The oscillating nature of sin and cos lead me to believe that sinx and cosx don't have limits at infinity. But if not, I don't know what to do. I want to try using L'hopitals but as far I can tell this isn't in an indeterminate form.
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    MHF Contributor FernandoRevilla's Avatar
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    Although \lim_{x\to +\infty}\sin x does not exist we have |\sin x|\leq 1 for all x\in \mathbb{R} . This means that \lim_{x\to +\infty}(x+\sin x)=+\infty . Same considerations for the denominator.
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    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by evankiefl View Post
    limit x---> infinity (x+sinx)/(3x+cosx)

    The oscillating nature of sin and cos lead me to believe that sinx and cosx don't have limits at infinity. But if not, I don't know what to do. I want to try using L'hopitals but as far I can tell this isn't in an indeterminate form.
    What You have to do is simply to devide numerator and denominator by x obtaining...

    \displaystyle \frac{x+ \sin x}{3x + \cos x} = \frac{1 + \frac{\sin x}{x}}{3+\frac{\cos x}{x}}

    ... and then find the limit if x tends to infinity...

    Kind regards

    \chi \sigma
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