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onegirl

  • 3 years ago

lim x->1 (sqrt(x) - 1)/ (x - 1)

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  1. hartnn
    • 3 years ago
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    hint : \((x-1)=(\sqrt x)^2-1^2 \) now use the difference of squares \(a^2-b^2=(a+b)(a-b)\)

  2. onegirl
    • 3 years ago
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    how do i do that?

  3. hartnn
    • 3 years ago
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    \((x-1)=(\sqrt x)^2-1^2=(\sqrt x-1)(\sqrt x+1)\) got this ? what gets cancelled ?

  4. onegirl
    • 3 years ago
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    okay the sqrt(x) - 1 gets cancelled

  5. hartnn
    • 3 years ago
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    yes, you can directly put x=1 now.

  6. onegirl
    • 3 years ago
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    okay so i substitute x = 1 into sqrt(x) - 1?

  7. hartnn
    • 3 years ago
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    no....that got cancelled....what remains ?

  8. onegirl
    • 3 years ago
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    what remains is (x - 1)

  9. hartnn
    • 3 years ago
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    but we wrote x-1 as \((x-1)=(\sqrt x)^2-1^2=(\sqrt x-1)(\sqrt x+1)\)

  10. onegirl
    • 3 years ago
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    ok

  11. hartnn
    • 3 years ago
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    \(\dfrac{\sqrt x-1}{x-1}=\dfrac{\sqrt{x}-1}{(\sqrt x-1 )(\sqrt x +1)}=\dfrac{1}{\sqrt x+1}\) got this ? now put x=1

  12. onegirl
    • 3 years ago
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    ok i get 1/2

  13. hartnn
    • 3 years ago
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    i get the same, its correct.

  14. onegirl
    • 3 years ago
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    ok i get 1/2

  15. hartnn
    • 3 years ago
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    yes, 1/2 is correct.

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