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Math Help - Not L'Hopital rule

  1. #1
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    Not L'Hopital rule

    limx->infinite ((9X+1)^1/2)/((X+1)^1/2)


    limx->90'- secX/tanx


    what it means by putting positive and negative behind?how to solve by using other way?
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  2. #2
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    Quote Originally Posted by Joyce View Post
    limx->infinite ((9X+1)^1/2)/((X+1)^1/2)


    ...
    Hi,

    \lim_{x \to \infty}{\frac{\sqrt{9x+1}}{\sqrt{x+1}}} = \lim_{x \to \infty}{\frac{3 \sqrt{x} \cdot \overbrace{\sqrt{1+\frac1{9x}}}^{\text{limit is 1}}}{\sqrt{x} \cdot \underbrace{\sqrt{1+\frac1x}}_{\text{limit is 1}}}} = 3
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  3. #3
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    Quote Originally Posted by Joyce View Post
    ...

    limx->90'- secX/tanx

    what it means by putting positive and negative behind?how to solve by using other way?
    Hi,

    \lim_{x \to \frac {\pi}{2}}\left(-\frac{\sec(x)}{\tan(x)}\right)=\lim_{x \to \frac {\pi}2}\left(-\frac{\frac{1}{\cos(x)}}{\frac{\sin(x)}{\cos(x)}}\  right)= \lim_{x \to \frac {\pi}2}\left(-\frac{1}{\cos(x)} \cdot \frac{\cos(x)}{\sin(x)}\right)=\lim_{x \to \frac {\pi}2}\left(-\frac{1}{\sin(x)}\right) = -1

    Quote Originally Posted by Joyce View Post
    what it means by putting positive and negative behind?how to solve by using other way?
    I only can guess because I don't know this way of writing: Probably the - or + sign indicates the direction of approach. So 90į- could mean that you approaches the 90į from the left where the values are smaller than 90į. Ask your teacher or try to find an explanation in your textbook.
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  4. #4
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    Hello, Joyce!

    Another approach . . .


    \lim_{x\to\infty}\frac{\sqrt{9x+1}}{\sqrt{x+2}}

    We have: . \lim_{x\to\infty}\sqrt{\frac{9x+1}{x+2}}

    Divide top and bottom by x\!:\;\;\lim_{x\to\infty}\sqrt{\frac{9 + \frac{1}{x}}{1 + \frac{2}{x}}} \;=\;\sqrt{\frac{9+0}{1+0}} \:=\:\sqrt{9} \;=\;3

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