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Math Help - solve for limit with square root in numerator and denominator

  1. #1
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    solve for limit with square root in numerator and denominator

    How would I solve a problem like this
    lim (sqrt(3x+3)-3)/(sqrt(10x+5)-5)
    x->2
    obviously the substitution method doesnt work so I was wondering if someone can help me with this one.
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by macro View Post
    How would I solve a problem like this
    lim (sqrt(3x+3)-3)/(sqrt(10x+5)-5)
    x->2
    obviously the substitution method doesnt work so I was wondering if someone can help me with this one.
    multiply the numerator and denominator of the given fraction by (sqrt(10x+5)-5), and you will have

    (sqrt(3x+3)-3)*(sqrt(10x+5)-5)
    lim -----------------------------------
    x->2 (sqrt(10x+5)-5)*(sqrt(10x+5)-5)

    NOTE that the denominator above is in the form of (a-b)(a-b), which is equal to a^2-2ab+b^2

    (sqrt(3x+3)-3)*(sqrt(10x+5)-5)
    = lim --------------------------------------
    x->2 (10x+5) - 2*5*(sqrt(10x+5)-5) + 25


    calculate the limit as x-> 2, you should get 0.
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  3. #3
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    It ends up with a zero in the denominator
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  4. #4
    Member Miss's Avatar
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    Quote Originally Posted by macro View Post
    How would I solve a problem like this
    lim (sqrt(3x+3)-3)/(sqrt(10x+5)-5)
    x->2
    obviously the substitution method doesnt work so I was wondering if someone can help me with this one.
    Hello.

    \lim_{x\to 2} \frac{ \sqrt{3x+3}-3 } { \sqrt{10x+5}-5 } = \sqrt{\frac{3}{5}} \, \lim_{x \to 2 } \frac{ \sqrt{x+1} - \sqrt{3} } { \sqrt{2x+1} - \sqrt{5} }

    Start by multiplying the function by \frac{\sqrt{x+1}+\sqrt{3}}{\sqrt{x+1}+\sqrt{3}}.
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  5. #5
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by macro View Post
    It ends up with a zero in the denominator
    How does it end up with a zero in the denominator?

    (sqrt(3x+3)-3)*(sqrt(10x+5)-5)
    = lim --------------------------------------
    x->2 (10x+5) - 2*5*(sqrt(10x+5)-5) + 25


    The denominator will be:

    (20+5) - 10*(sqrt(25)-5) +25

    = 25 -10*0 +25
    = 50

    ???
    The numerator should be a 0 here, and the overall function goes to 0 as x tends to 2!
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