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Question

If $f left parenthesis x right parenthesis equals x t a n left parenthesis negative 1 right parenthesis invisible function application x$ then $l i m subscript left parenthesis x rightwards arrow 1 right parenthesis left parenthesis f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis equals$

$π / 4$
$(π + 1) / 4$
$(π + 2) / 4$
$(π + 3) / 4$

hint Hint:

In this question we are getting $0 over 0$ form. We will use the L'Hôpital's rule to find the value of the limit

The correct answer is: $left parenthesis pi plus 2 right parenthesis divided by 4$

In this question we are $f left parenthesis x right parenthesis equals tan to the power of negative 1 end exponent open parentheses x close parentheses$ and we have to find the value of $limit as x minus greater than 1 of fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction$
Step1: Putting the value of limit
By putting the value of limit we are getting both numerator and denominator as zero. That is $0 over 0$ form.
Step2: Using L'Hôpital's rule
By differentiating both numerator and denominator we get
$limit as x minus greater than 1 of fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 1 end fraction$
=> $f apostrophe left parenthesis 1 right parenthesis$
We have to find the value of $f apostrophe left parenthesis 1 right parenthesis$
Step3: Finding the value of $f apostrophe left parenthesis 1 right parenthesis$ .
$f left parenthesis x right parenthesis equals tan to the power of negative 1 end exponent open parentheses x close parentheses$
=> $f apostrophe left parenthesis x right parenthesis equals tan to the power of negative 1 end exponent open parentheses x close parentheses plus fraction numerator x over denominator 1 plus x squared end fraction$
by putting the value of $x equals 1$ ,
$f apostrophe left parenthesis 1 right parenthesis space equals space pi over 4 plus 1 half$
=> $fraction numerator pi plus 2 over denominator 4 end fraction$

Related Questions to study

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 $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 2 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis$

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 $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 2 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis$

$L t left parenthesis n rightwards arrow infinity right parenthesis left square bracket 1 divided by n squared s e c squared 1 divided by n squared plus 2 divided by n squared s e c squared 4 divided by n squared plus midline horizontal ellipsis. plus 1 divided by n s e c squared 1 right square bracket equals$

$L t left parenthesis n rightwards arrow infinity right parenthesis left square bracket 1 divided by n squared s e c squared 1 divided by n squared plus 2 divided by n squared s e c squared 4 divided by n squared plus midline horizontal ellipsis. plus 1 divided by n s e c squared 1 right square bracket equals$

$L t left parenthesis n rightwards arrow infinity right parenthesis left parenthesis 1 plus 2 to the power of 4 plus 3 to the power of 4 plus midline horizontal ellipsis plus n to the power of 4 right parenthesis divided by n to the power of 5 minus L t left parenthesis 1 plus 2 cubed plus 3 cubed plus.. plus n cubed right parenthesis divided by n to the power of 4 equals$

$L t left parenthesis n rightwards arrow infinity right parenthesis left parenthesis 1 plus 2 to the power of 4 plus 3 to the power of 4 plus midline horizontal ellipsis plus n to the power of 4 right parenthesis divided by n to the power of 5 minus L t left parenthesis 1 plus 2 cubed plus 3 cubed plus.. plus n cubed right parenthesis divided by n to the power of 4 equals$

parallel

If $f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis plus x comma y space$ and $f left parenthesis 5 right parenthesis equals 2 comma f to the power of apostrophe left parenthesis 0 right parenthesis equals 3$ then $f to the power of apostrophe left parenthesis 5 right parenthesis$ is

If $f left parenthesis x plus y right parenthesis equals f left parenthesis x right parenthesis f left parenthesis y right parenthesis plus x comma y space$ and $f left parenthesis 5 right parenthesis equals 2 comma f to the power of apostrophe left parenthesis 0 right parenthesis equals 3$ then $f to the power of apostrophe left parenthesis 5 right parenthesis$ is

$L t ┬ left parenthesis n rightwards arrow infinity right parenthesis left parenthesis 1 to the power of p plus 2 to the power of p plus 3 to the power of p plus midline horizontal ellipsis. plus n to the power of p right parenthesis divided by n to the power of left parenthesis p plus 1 right parenthesis$ is

$L t ┬ left parenthesis n rightwards arrow infinity right parenthesis left parenthesis 1 to the power of p plus 2 to the power of p plus 3 to the power of p plus midline horizontal ellipsis. plus n to the power of p right parenthesis divided by n to the power of left parenthesis p plus 1 right parenthesis$ is

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

parallel

$L t left parenthesis h rightwards arrow 0 right parenthesis left parenthesis f left parenthesis 2 h plus 2 plus h squared right parenthesis minus f left parenthesis 2 right parenthesis right parenthesis divided by left parenthesis f left parenthesis h minus h squared plus 1 right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis$ , given that $f to the power of apostrophe left parenthesis 2 right parenthesis equals 6$ and $f to the power of apostrophe left parenthesis 1 right parenthesis equals 4$

$L t left parenthesis h rightwards arrow 0 right parenthesis left parenthesis f left parenthesis 2 h plus 2 plus h squared right parenthesis minus f left parenthesis 2 right parenthesis right parenthesis divided by left parenthesis f left parenthesis h minus h squared plus 1 right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis$ , given that $f to the power of apostrophe left parenthesis 2 right parenthesis equals 6$ and $f to the power of apostrophe left parenthesis 1 right parenthesis equals 4$

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

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

If $f left parenthesis a right parenthesis equals 2 comma f to the power of apostrophe left parenthesis a right parenthesis equals 1 comma g left parenthesis a right parenthesis equals negative 1 comma g to the power of factorial left parenthesis a right parenthesis equals 2 comma$ then the value of $L i m subscript left parenthesis x rightwards arrow a right parenthesis invisible function application left parenthesis g left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f left parenthesis x right parenthesis right parenthesis divided by left parenthesis x minus a right parenthesis$ is
,

If $f left parenthesis a right parenthesis equals 2 comma f to the power of apostrophe left parenthesis a right parenthesis equals 1 comma g left parenthesis a right parenthesis equals negative 1 comma g to the power of factorial left parenthesis a right parenthesis equals 2 comma$ then the value of $L i m subscript left parenthesis x rightwards arrow a right parenthesis invisible function application left parenthesis g left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f left parenthesis x right parenthesis right parenthesis divided by left parenthesis x minus a right parenthesis$ is
,

parallel

If $G left parenthesis x right parenthesis equals negative √ left parenthesis 25 minus x squared right parenthesis$ then $L i m subscript left parenthesis x rightwards arrow 1 right parenthesis invisible function application left parenthesis G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis$ has the value

If $G left parenthesis x right parenthesis equals negative √ left parenthesis 25 minus x squared right parenthesis$ then $L i m subscript left parenthesis x rightwards arrow 1 right parenthesis invisible function application left parenthesis G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis$ has the value

If $f left parenthesis x right parenthesis equals √ left parenthesis left parenthesis x minus sin invisible function application x right parenthesis divided by left parenthesis x plus cos invisible function application x right parenthesis right parenthesis$ then $L t f left parenthesis x right parenthesis space$ is

If $f left parenthesis x right parenthesis equals √ left parenthesis left parenthesis x minus sin invisible function application x right parenthesis divided by left parenthesis x plus cos invisible function application x right parenthesis right parenthesis$ then $L t f left parenthesis x right parenthesis space$ is

If $f left parenthesis 9 right parenthesis equals 9 comma f to the power of 1 left parenthesis 9 right parenthesis minus 4 comma$ then $L t divided by left parenthesis x rightwards arrow t right parenthesis space space left parenthesis √ left parenthesis f left parenthesis x right parenthesis right parenthesis minus 3 right parenthesis divided by left parenthesis √ x minus 3 right parenthesis$

If $f left parenthesis 9 right parenthesis equals 9 comma f to the power of 1 left parenthesis 9 right parenthesis minus 4 comma$ then $L t divided by left parenthesis x rightwards arrow t right parenthesis space space left parenthesis √ left parenthesis f left parenthesis x right parenthesis right parenthesis minus 3 right parenthesis divided by left parenthesis √ x minus 3 right parenthesis$

parallel

$L t subscript left parenthesis x rightwards arrow negative infinity right parenthesis left square bracket left parenthesis x to the power of 4 s i n invisible function application left parenthesis 1 divided by x right parenthesis plus x squared right parenthesis divided by left parenthesis left parenthesis 1 plus vertical line x vertical line cubed right parenthesis right parenthesis right square bracket equals$

$L t subscript left parenthesis x rightwards arrow negative infinity right parenthesis left square bracket left parenthesis x to the power of 4 s i n invisible function application left parenthesis 1 divided by x right parenthesis plus x squared right parenthesis divided by left parenthesis left parenthesis 1 plus vertical line x vertical line cubed right parenthesis right parenthesis right square bracket equals$

$L t subscript left parenthesis x rightwards arrow infinity right parenthesis invisible function application left square bracket left parenthesis a to the power of left parenthesis 1 divided by x right parenthesis plus b to the power of left parenthesis 1 divided by x right parenthesis plus c to the power of left parenthesis 1 divided by x right parenthesis right parenthesis divided by 3 right square bracket to the power of x$ where a,b,c are real and non-zero=

$L t subscript left parenthesis x rightwards arrow infinity right parenthesis invisible function application left square bracket left parenthesis a to the power of left parenthesis 1 divided by x right parenthesis plus b to the power of left parenthesis 1 divided by x right parenthesis plus c to the power of left parenthesis 1 divided by x right parenthesis right parenthesis divided by 3 right square bracket to the power of x$ where a,b,c are real and non-zero=

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

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

parallel

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