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Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim x → ∞ x5/ √(x10 + 7)?

6 Answers

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  • ?
    Lv 7
    2 months ago

    I understand your question as

    limit [(x^5)/sqr(x^10+7)]=? If so, then

    x->inf.

    the LHS=

    limit [(x^5)/{(x^5)sqr[1+7/(x^10)]}=1.

    x->inf.

  • 2 months ago

    lim ........x^5

    x->∞ ------------

    ........√(x^10 + 7)

    ...........x^5/x^5

    =lim ------------

    x->∞ √(x^10/x^10 + 7/x^10)

    .............1

    =lim -------------

    x->∞...√(1)

    = 1/1

    = 1 answer//

  • ?
    Lv 6
    2 months ago

    [lim x→inf] x^5 / √(x^10 + 7) ; we know x is positive

    = [lim x→inf] [x^5/x^5] / [√(x^10 + 7)/x^5]

    = [lim x→inf] 1 / [√((x^10 + 7)/x^10)]

    = [lim x→inf] 1 / [√(1 + 7/x^10)]

    = 1 ; because 7/x^10 approaches 0 when x goes to inf.

  • ?
    Lv 7
    2 months ago

    Factor out x5 from top & bottom to get:

    (x5/x5) * 1 / √( 1 + 7/x10) = 

    1 / √( 1 + 7/x10).

    This is much simpler, no? 

    As x --> ∞ this gives? 

    Done!

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  • 2 months ago

    You can use the "sandwich method"

    a) remember that on the way to a limit, as x->∞, you use large positive integers, for x

    b) if you make the denominator larger, then the whole fraction is smaller (and vice-versa: a smaller denominator means a larger fraction.

    ---

    x^5 / √(x^10 + 7)

    The denominator is √(x^10 + 7)

    x^10 < x^10 + 7

    Therefore, the fraction x^5 / √x^10

    is larger than the original fraction (it has a smaller denominator)

    Lim x->∞ { x^5 / √x^10 } =

    Lim x->∞ { x^5 / x^5 } = 1

    The limit of the original fraction cannot be larger than 1.

    ---

    Since x is positive, adding x values in the denominator can only make it larger;

    larger denominator = smaller fraction

    √(x^10 + 8x^5 + 16) > √(x^10 + 7)

    √[(x^5 + 4)^2]  > √(x^10 + 7)

    (x^5 + 4) > √(x^10 + 7)

    Therefore, the fraction x^5 / (x^5 + 4) is smaller than the original fraction

    The limit of that fraction (as x goes to infinity) is 1

    The limit of the original fraction cannot be smaller than 1.

    ----

    Cannot be bigger that 1

    Cannot be smaller than 1

    Then it must be 1

  • 2 months ago

    The customary way to write an exponent using a typical keyboard is with the caret " ^ " -- like, x^2 for "x squared" and x^3 for "x cubed."

    So I'll assume you meant

    lim (x --> infinity) [x^5 / sqrt(x^10 + 7)].

    The limit is 1, because when x is very large, the square root of x^10 becomes nearly indistinguishable from x^5.

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