Solution for the question - if f:r rarr r be a positive increasing function with (t(x rarroo)(f(3x))//(f(x))=1 then f(x rarr oo)(f(2x))//f(x)) 33//2 '/> 2//3 '/>

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Question

If $f colon R rightwards arrow R$ be a positive increasing function with $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 3 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis equals 1 space$ then $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 2 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis$

1
$2 / 3$
$3 / 2$
3

hint Hint:

In this question we are given $f left parenthesis x right parenthesis$ is an increasing for all $x greater than 0$ . Then we will use sandwich theorem to find the value of the limit

The correct answer is: 1

In this question we are given $limit as x rightwards arrow infinity of equals fraction numerator f left parenthesis 3 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction equals 1$ and we have to find the value of $limit as x rightwards arrow infinity of fraction numerator f left parenthesis 2 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction$
Step1: Using Sandwich Theorem
In the question we are given $f left parenthesis x right parenthesis$ is an increasing for all $x greater than 0$ . Then
$f left parenthesis x right parenthesis less or equal than f left parenthesis 2 x right parenthesis less or equal than f left parenthesis 3 x right parenthesis$
By dividing by $f left parenthesis x right parenthesis$ we get,
=> $fraction numerator f left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis end fraction less or equal than fraction numerator f left parenthesis 2 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction less or equal than fraction numerator f left parenthesis 3 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction$
By taking the limit $limit as x rightwards arrow infinity of$ we get
=> $limit as x rightwards arrow infinity of fraction numerator f left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis end fraction less or equal than limit as x rightwards arrow infinity of fraction numerator f left parenthesis 2 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction less or equal than limit as x rightwards arrow infinity of fraction numerator f left parenthesis 3 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction$
Given $limit as x rightwards arrow infinity of equals fraction numerator f left parenthesis 3 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction equals 1$ and we can see that $limit as x rightwards arrow infinity of equals fraction numerator f left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis end fraction equals 1$
=> $1 less or equal than limit as x rightwards arrow infinity of fraction numerator f left parenthesis 2 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction less or equal than 1$
So, the value of the $limit as x rightwards arrow infinity of fraction numerator f left parenthesis 2 x right parenthesis over denominator f left parenthesis x right parenthesis end fraction$ is $1$

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1
$2 / 3$
$3 / 2$
3

In this question we are given $f left parenthesis x right parenthesis$ is an increasing for all $x greater than 0$ . Then we will use sandwich theorem to find the value of the limit

The correct answer is: 1

Related Questions to study

For $x element of R L t left parenthesis x rightwards arrow infinity right parenthesis left parenthesis left parenthesis x minus 3 right parenthesis divided by left parenthesis x plus 2 right parenthesis right parenthesis to the power of x equals$

For $x element of R L t left parenthesis x rightwards arrow infinity right parenthesis left parenthesis left parenthesis x minus 3 right parenthesis divided by left parenthesis x plus 2 right parenthesis right parenthesis to the power of x equals$

$L i m left parenthesis x rightwards arrow 0 right parenthesis left parenthesis x t a n 2 x minus 2 x t a n x right parenthesis divided by left parenthesis left parenthesis 1 minus c o s invisible function application 2 x right parenthesis squared right parenthesis equals$

$L i m left parenthesis x rightwards arrow 0 right parenthesis left parenthesis x t a n 2 x minus 2 x t a n x right parenthesis divided by left parenthesis left parenthesis 1 minus c o s invisible function application 2 x right parenthesis squared right parenthesis equals$

$L t subscript left parenthesis x rightwards arrow infinity right parenthesis invisible function application left parenthesis left parenthesis x squared plus 5 x plus 3 right parenthesis divided by left parenthesis x squared plus x plus 2 right parenthesis right parenthesis to the power of x equals$

$L t subscript left parenthesis x rightwards arrow infinity right parenthesis invisible function application left parenthesis left parenthesis x squared plus 5 x plus 3 right parenthesis divided by left parenthesis x squared plus x plus 2 right parenthesis right parenthesis to the power of x equals$

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If be a positive increasing function with then

13

In this question we are given is an increasing for all . Then we will use sandwich theorem to find the value of the limit

The correct answer is: 1

In this question we are given and we have to find the value of Step1: Using Sandwich TheoremIn the question we are given is an increasing for all . ThenBy dividing by we get,=> By taking the limit we get=> Given and we can see that =>So, the value of the is

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where a,b,c are real and non-zero=

where a,b,c are real and non-zero=

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1
$2 / 3$
$3 / 2$
3

In this question we are given $f left parenthesis x right parenthesis$ is an increasing for all $x greater than 0$ . Then we will use sandwich theorem to find the value of the limit

For $x element of R L t left parenthesis x rightwards arrow infinity right parenthesis left parenthesis left parenthesis x minus 3 right parenthesis divided by left parenthesis x plus 2 right parenthesis right parenthesis to the power of x equals$

For $x element of R L t left parenthesis x rightwards arrow infinity right parenthesis left parenthesis left parenthesis x minus 3 right parenthesis divided by left parenthesis x plus 2 right parenthesis right parenthesis to the power of x equals$