Question

How I resolve this limit?

`\lim _ ( x \rightarrow 7 ) (root3(1+x)-root3(2x-6))/( sqrt( 7 x ) - sqrt( 7 + 6 x ) )`

Answer 1

`- 7/6`

Explanation:

`\lim _ ( x \rightarrow 7 ) (root3(1+x)-root3(2x-6))/( sqrt( 7 x ) - sqrt( 7 + 6 x ) )`

Let: `y = x-7 qquad qquad x = y + 7`

`= lim _ ( y to bb 0 ) (root3(y+8)-root3(2y+8))/( sqrt( 7y + 49 ) - sqrt( 49 + 6y ) )`

`lim _ ( y to 0 ) (2 root3(1+y/8)- 2root3(1+y/4))/( 7 sqrt( 1+y/7 ) - 7sqrt( 1 + (6y)/49 ) )`

With Binomial Expansions:

`= lim _ ( y to 0 ) (2 (1+1/3* y/8 + bbbO(y^2) )- 2 (1+1/3* y/4 + bbbO(y^2)))/( 7 ( 1+1/2*y/7 +bbbO(y^2) ) - 7 ( 1 + 1/2* (6y)/49 + bbbO(y^2)) )`

`= 2/7 lim _ ( y to 0 ) ( y/24 - y/12 + bbbO(y^2) )/( y/14 - (3y)/49+ bbbO(y^2))`

`= 2/7 lim _ ( y to 0 ) ( -1/24 + bbbO(y) )/( 1/98+ bbbO(y))`

`= - 2/7 * 98/24 = - 7/6`

Answer 2

`lim_(xto7)(root(3)(1+x)-root(3)(2x-6))/(sqrt(7x)-sqrt(7+6x))=-7/6`

Explanation:

@Ultrilliam wrote an answer with the help of

Binomial Expansion. This method is short.

I have tried to solve with the help of formula given below.

This method need some adjustment for `A`

`(1)lim_(xtoa) (x^n-a^n)/(x-a)=n*a^(n-1) orcolor(red)( lim_(XtoA) (X^n-A^n)/(X-A)=n*A^(n-1)`
........................................................................................................
Let,

`L=lim_(xto7)(root(3)(1+x)-root(3)(2x-6))/(sqrt(7x)-sqrt(7+6x))`

#=lim_(xto7)(root(3)(1+x)-root(3)(2x-6))/(sqrt(7x)- sqrt(7+6x))xxcolor(blue) ((sqrt(7x)+sqrt(7+6x))/(sqrt(7x)+sqrt(7+6x))#

#=lim_(xto7)(root(3)(1+x)-root(3)(2x-6))/((7x)- (7+6x))xxcolor(blue)((sqrt(7x)+sqrt(7+6x))#

#=lim_(xto7)((1+x)^(1/3)-color(violet)((8)^(1/3))-(2x-6)^(1/3)+color(violet)((8)^(1/3)))/(x- 7)xx{sqrt(49)+sqrt(49)}#

#=14{lim_(xto7)((1+x)^(1/3)-(8)^(1/3))/(x-7)-lim_(xto7)((2x- 6)^(1/3)-(8)^(1/3))/(x-7)}#

#=14{color(red)(lim_(xto7)((1+x)^(1/3)-(8)^(1/3))/((1+x)-(8))-lim_(xto7)((2x- 6)^(1/3)-(8)^(1/3))/((2x-6)-(8)))xx2}#

`=14{color(red)(1/3*(8)^(1/3-1)-1/3*(8)^(1/3-1))xx2}...tocolor(red)(Apply(1)`

`=14/3{(2^3)^(-2/3)-(2^3)^(-2/3)xx2}`

`=14/3{2^(-2)-2*2^(-2)}`

`=14/3{1/4-2/4}`

`=14/3*(-1/4)`

`=-7/6`