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calculus

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Can someone please help me with these two? im so lost and i really appreciate it if someone could show me what to do!

1) To evaluate
lim x→infinity sqrt (x^2 + 4)
first consider that as x becomes infinitely large, x^2+4 -> _____



2)Find lim x→ infinity f(x)if,
for all x > 1,

(6e^x − 17)/3e^x < f(x) < 2sqrt(x)/ sqrt (x − 1)

  • calculus - ,

    forget the first one i got it. just the second one i don't know where to begin

  • calculus - ,

    for (6e^x - 17)/(3e^x)
    divide top and bottom by e^x
    we get
    (6-17/e^x)/3 , as x ----> ∞ , the expression ---> 2

    and 2√x/√(x-1) ----> 2 , as x ---->∞

    so as x---->∞
    2 < f(x) < 2
    which implies f(x) = 2

  • calculus - ,

    SOOOO HAPPY YOU ANSWERED ME!
    ok now im just trying to process this in my head. i understand dividing it by e^x but you lost me at the expression -->2
    sorry to be a pain in the ass, i just want to make sure i fully understand this. thank you so much! you are a life saver!

  • calculus - ,

    in (6-17/e^x)/3 , as x ----> ∞
    what happens when you divide 17 by a very very very large number ?
    Doesn't it go to zero ?
    so you have (6-0)/3 = 2

    for the right side 2√x/√(x-1)

    when x is really really large, do you think it makes any difference if we take √x or √(x-1) ?
    e.g. take
    √123456789 and √123456788
    now divide them to get 1.000000004
    so the right side would be 2.00000008 or just plain 2

    so the f(x) is sandwiched between 2 and 2 , thus it MUST BE 2

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