If f(9)=9 and f^'(9)=4, then what is the value of lim_(x to9)(sqrt(f(x))-3)/(sqrtx -3)?

1 Answer
Jun 15, 2018

lim_(xto9) (sqrtf(x)-3)/(sqrtx-3)=4

Explanation:

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