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Math Help - Basic Limits

  1. #1
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    Basic Limits

    Hi, I need a hand evaluating a couple of limits without using L'hopital's.
    I've done a ton of these so far and I often need to use little 'tricks' like factorising/cancelling etc, but I can't see these two.

     \frac{\sqrt{x+2}-3}{x-7} as x approaches 7

    and

     \frac{\frac{1}{4} + \frac{1}{x}}{4+x} as x approaches -4.

    Cheers.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by U-God View Post
    Hi, I need a hand evaluating a couple of limits without using L'hopital's.
    I've done a ton of these so far and I often need to use little 'tricks' like factorising/cancelling etc, but I can't see these two.

     \frac{\sqrt{x+2}-3}{x-7} as x approaches 7
    Multiply by the conjugate of the numerator:

    \lim_{x\to7}\frac{\sqrt{x+2}-3}{x-7}=\lim_{x\to 7}\frac{\sqrt{x+2}-3}{x-7}\cdot\frac{\sqrt{x+2}+3}{\sqrt{x+2}+3}=\lim_{x\t  o7}\frac{x+2-9}{(x-7)(\sqrt{x+2}+3)}

    Can you take it from here?

    and

     \frac{\frac{1}{4} + \frac{1}{x}}{4+x} as x approaches -4.

    Cheers.
    Combine the terms in the numerator. The LCD is 4x:


    \lim_{x\to -4}\frac{\frac{x}{4x} + \frac{4}{4x}}{4+x}=\lim_{x\to-4} \frac{\frac{4+x}{4x}}{4+x}

    Can you take it from here?

    --Chris
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    Multiply by the conjugate of the numerator:

    \lim_{x\to7}\frac{\sqrt{x+2}-3}{x-7}=\lim_{x\to 7}\frac{\sqrt{x+2}-3}{x-7}\cdot\frac{\sqrt{x+2}+3}{\sqrt{x+2}+3}=\lim_{x\t  o7}\frac{x+2-9}{(x-7)(\sqrt{x+2}+3)}

    Can you take it from here?



    Combine the terms in the numerator. The LCD is 4x:


    \lim_{x\to -4}\frac{\frac{x}{4x} + \frac{4}{4x}}{4+x}=\lim_{x\to-4} \frac{\frac{4+x}{4x}}{4+x}

    Can you take it from here?

    --Chris
    I certainly can take it from there, thanks tons
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  4. #4
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    Quote Originally Posted by U-God View Post
    Hi, I need a hand evaluating a couple of limits without using L'hopital's.
    I've done a ton of these so far and I often need to use little 'tricks' like factorising/cancelling etc, but I can't see these two.

     \frac{\sqrt{x+2}-3}{x-7} as x approaches 7

    and

     \frac{\frac{1}{4} + \frac{1}{x}}{4+x} as x approaches -4.

    Cheers.
    For the first, Take the derivative of the top and the derivative of the bottom, then sub 7 into both.
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  5. #5
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    Quote Originally Posted by Prove It View Post
    For the first, Take the derivative of the top and the derivative of the bottom, then sub 7 into both.
    Thanks, but I can't use l'hopital's rule
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