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Thread: Some limit questions

  1. #1
    Member Vinod's Avatar
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    Some limit questions

    Let I be an open interval such that 4 ∈ I and let a function f be defined on
    a set D = I\{4}. Evaluate $ \lim_{x \to 4} f(x)$ , where x + 2 ≤ f(x) ≤ x^2 − 10 for all
    x ∈ D.

    2)How to use the squeeze theorem to show that $ \lim_{x\to0^+\left(\sqrt{x}e^{sin\frac1x}\right)}= 0$
    Last edited by topsquark; Apr 11th 2018 at 10:33 PM.
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  2. #2
    MHF Contributor

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    Re: Some limit questions

    Quote Originally Posted by Vinod View Post
    Let I be an open interval such that 4 ∈ I and let a function f be defined on
    a set D = I\{4}. Evaluate $ \lim_{x \to 4} f(x)$ , where x + 2 ≤ f(x) ≤ x^2 − 10 for all
    x ∈ D.
    It is pretty obvious, isn't it, that for x= 4, the given equality becomes $\displaystyle 6\le f(4)\le 6$. If you are concerned about the fact that 4 is NOT in D, don't be. As long as there are points in D arbitrarily close to 4 we can take the limit as x goes to 4.

    2)How to use the squeeze theorem to show that $ \lim_{x\to0^+\left(\sqrt{x}e^{sin\frac1x}\right)}= 0$
    Use the fact that $\displaystyle e^x$ is an increasing function and that $\displaystyle sin(1/x)$ lies between -1 and 1.
    Last edited by HallsofIvy; Apr 12th 2018 at 04:31 AM.
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