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Question

$L t left parenthesis x rightwards arrow pi divided by 6 right parenthesis left parenthesis left parenthesis 3 s i n x minus √ 3 c o s x right parenthesis divided by left parenthesis 6 x minus pi right parenthesis right parenthesis equals$

$\sqrt 3$
$1 / \sqrt 3$
$- \sqrt 3$
$(- 1) / \sqrt 3$

hint Hint:

In this question we have to rearrange the expression and then use the standard limits $limit as x minus greater than 0 of fraction numerator sin open parentheses x close parentheses over denominator x end fraction equals 1$ .

The correct answer is: $space 1 divided by √ 3$

In this question we have to find the limit. $limit as x minus greater than pi over 6 of open parentheses fraction numerator 3 sin open parentheses x close parentheses minus square root of 3 cos open parentheses x close parentheses over denominator 6 x minus straight pi end fraction close parentheses$
Step1: Rearranging the expression
$limit as x minus greater than pi over 6 of open parentheses fraction numerator square root of 3 left parenthesis square root of 3 sin open parentheses x close parentheses minus cos open parentheses x close parentheses right parenthesis over denominator 6 left parenthesis x minus begin display style straight pi over 6 end style right parenthesis end fraction close parentheses$
By multiplying and dividing by $2$ , we get,
$limit as x minus greater than pi over 6 of open parentheses fraction numerator 2 square root of 3 left parenthesis begin display style fraction numerator square root of 3 over denominator 2 end fraction end style sin open parentheses x close parentheses minus begin display style 1 half end style cos open parentheses x close parentheses right parenthesis over denominator 6 left parenthesis x minus begin display style straight pi over 6 end style right parenthesis end fraction close parentheses$
=> $limit as x minus greater than pi over 6 of fraction numerator 1 over denominator square root of 3 end fraction open parentheses fraction numerator left parenthesis begin display style cos open parentheses straight pi over 6 close parentheses end style sin open parentheses x close parentheses minus begin display style sin open parentheses straight pi over 6 close parentheses end style cos open parentheses x close parentheses right parenthesis over denominator left parenthesis x minus begin display style straight pi over 6 end style right parenthesis end fraction close parentheses$
=> $limit as x minus greater than pi over 6 of fraction numerator 1 over denominator square root of 3 end fraction open parentheses fraction numerator sin open parentheses x minus begin display style straight pi over 6 end style close parentheses right parenthesis over denominator left parenthesis x minus begin display style straight pi over 6 end style right parenthesis end fraction close parentheses$
Step2: Using the standard limits
We know that $limit as x minus greater than 0 of fraction numerator sin open parentheses x close parentheses over denominator x end fraction equals 1$
=. $limit as x minus greater than pi over 6 of fraction numerator 1 over denominator square root of 3 end fraction open parentheses fraction numerator sin open parentheses x minus begin display style straight pi over 6 end style close parentheses right parenthesis over denominator left parenthesis x minus begin display style straight pi over 6 end style right parenthesis end fraction close parentheses$
=> $fraction numerator 1 over denominator square root of 3 end fraction$

Related Questions to study

If $L t left parenthesis x rightwards arrow 0 right parenthesis left parenthesis left parenthesis c o s invisible function application 4 x plus a c o s invisible function application 2 x plus b right parenthesis divided by x to the power of 4 right parenthesis$ is finite then the values of a,b are respectively

If $L t left parenthesis x rightwards arrow 0 right parenthesis left parenthesis left parenthesis c o s invisible function application 4 x plus a c o s invisible function application 2 x plus b right parenthesis divided by x to the power of 4 right parenthesis$ is finite then the values of a,b are respectively

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

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

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

parallel

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

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

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

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

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

parallel

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

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

parallel

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

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

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
,

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

parallel

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 $L t divided by left parenthesis x rightwards arrow t right parenthesis space space left parenthesis √ left parenthesis f left parenthesis x right parenthesis right parenthesis minus 3 right parenthesis divided by left parenthesis √ x minus 3 right parenthesis$

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

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

parallel

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