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Let f :RtoR be a positive, increasing fu...

Let `f :RtoR` be a positive, increasing function with
`lim_(xtooo) (f(3x))/(f(x))=1`. Then `lim_(xtooo) (f(2x))/(f(x))` is equal to

Text Solution

Verified by Experts

The correct Answer is:
B
`f(x)` is a positive increasing function. Therefore,
`0ltf(x)ltf(2x)ltf(3x)`
`implies" "0lt1lt(f(2x))/(f(x))lt(f(3x))/(f(x))`
`implies" "underset(xtooo)lim1ltunderset(xtooo)lim(f(2x))/(f(x))ltunderset(xtooo)lim(f(3x))/(f(x))" "(becauseunderset(xtooo)lim(f(3x))/(f(x))=1)`
By Sandwich theorem, we get
`underset(xtooo)lim(f(2x))/(f(x))=1`
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Knowledge Check

  • If lim_(xto)(f(x))/(x^(2))=k then lim_(xto1)f(x)=

    A
    0
    B
    1
    C
    k
    D
    not defined

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