We know that,
(1)f'(c)=lim_(xtoc) (f(x)-f(c))/(x-c) , take x=9
=>f'(9)=lim_(xto9) (f(x)-f(9))/(x-9) ,where, f(9)=9
=>f'(9)=lim_(xto9)(f(x)-9)/(x-9) ,where, f'(9)=4
=>color(brown)(lim_(xto9)(f(x)-9)/(x-9)=4.....(2)
We take,
L=lim_(xto9) (sqrtf(x)-3)/(sqrtx-3)
color(white)(L)=lim_(xto9)(sqrtf(x)-3)/(sqrtx-3)xxcolor(red)
((1))xxcolor(blue)((1))
color(white)(L)=lim_(xto9)(sqrtf(x)-3)/(sqrtx-3)xxcolor(red)((sqrtf(x)+3)/(sqrtf(x)+3))xxcolor(blue)((sqrtx+3)/(sqrtx+3))
color(white)(L)=lim_(xto9)((sqrtf(x))^2-(3)^2)/((sqrtx)^2-(3)^2)xx(color(blue)(sqrtx+3))/(color(red)(sqrtf(x)+3))
color(white)(L)=color(brown)(lim_(xto9)(f(x)-9)/(x-9))xxlim_(xto9)(sqrtx+3)/(sqrtf(x)+3)...tocolor(brown)([use (2)]
color(white)(L)=color(brown)(4)xx(sqrt9+3)/(sqrtf(9)+3)
color(white)(L)=4[(3+3)/(sqrt9+3)]
:.L=4{6/6}=4