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 - ,

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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