lim h–>0 [(f(x+h)-f(x))/h]

x=9

lim h–>0 [(f(9+h))-f(9))/h]

= (((9+h)^2-sqrt(9+h))-

(9^2-sqrt(9))/h)

= (81+18h+h^2-sqrt(9+h)-81+3)/h

=(3-sqrt(9+h)+18h+h^2)/h

Group together the h terms and seperate the numerator

((h^2+18h)/h)+((3-sqrt(9+h))/h)

Now we have 2 seperate terms. We can take the limits individually and then add the results

1. lim h–>0 (h^2+18h)/h

and

2. lim h–>0 (3-sqrt(9+h))/h

1. lim h–>0 (h+18) (I factored and cancelled)

=18

2. lim h–>0 (3-sqrt(9+h))/h

For 2, multiply by its conjugate, 3+sqrt(9+h)

2. lim h–>0 (9-9-h)/(h(3+sqrt(9+h))

= lim h–>0 (-h/(h(3+sqrt(9+h))

= lim h–>0 (-1/(3+sqrt(9+h))

= -1/(3+3)

= -1/6

Add this to 18 and we obtain 107/6. This is the slope of the tangent line at x=9. Using the point-slope formula for a line we obtain the equation of the tangent line:

y-78=(107/6)(x-9)

You can rearrange to the slope-intercept form from here if you need to, but this answer is correct as it is.