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Math Help - Assistance with Evaluating Limits

  1. #1
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    Assistance with Evaluating Limits

    Hello, and thanks for the help in advance.

    I have two limits I must evaluate on a practice exam, and am having a little trouble. They are both complicated enough that you cannot simply plug in the value the variable is approaching. They are listed below.

    limit sin(4x) As x -> 0
    .........2x

    limit 1 - cos(t) As t -> 0
    .........t^2

    (The periods are simply so that the fractions look right.)

    I know you can pull 1/2 out in front of the limit for the first example (with x), but other than that I am at a loss as to what to do. I can do more complicated things...it's just been a while since I had to evaluate limits like this. Maybe it's a few simple things I'm forgetting?

    Thanks again, I hope the way I've typed them is legible.
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  2. #2
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    Quote Originally Posted by NBrunk View Post
    Hello, and thanks for the help in advance.

    I have two limits I must evaluate on a practice exam, and am having a little trouble. They are both complicated enough that you cannot simply plug in the value the variable is approaching. They are listed below.

    limit sin(4x) As x -> 0
    .........2x

    \color{red}\mbox{If you already know that } \frac{\sin x}{x} \xrightarrow [x \to 0] {} 1 \mbox{ , then it follows that } \frac{\sin f(x)}{f(x)} \xrightarrow [x \to 0] {} 1

    \color{red}\mbox{for any function }f(x)\,\mbox{ s.t. }f(x) \xrightarrow [x \to 0] {} 0\,\,when\,\, x\rightarrow 0\,\,and\,\,\sin f(x)\,\,exists

    \color{red}\mbox{Well, now just write }\frac{\sin 4x}{2x}=2\cdot \frac{\sin 4x}{4x}....



    limit 1 - cos(t) As t -> 0
    .........t^2


    \color{red}1-\cos t=2\sin^2(t/2)\Longrightarrow\,\frac{1-\cos t}{t^2}=\frac{1}{2}\cdot\left(\frac{\sin \frac{t}{2}}{\frac{t}{2}}\right)^2 ...

    \color{blue}Tonio


    (The periods are simply so that the fractions look right.)

    I know you can pull 1/2 out in front of the limit for the first example (with x), but other than that I am at a loss as to what to do. I can do more complicated things...it's just been a while since I had to evaluate limits like this. Maybe it's a few simple things I'm forgetting?

    Thanks again, I hope the way I've typed them is legible.
    .
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