Find the following limit without l'hop rule

lim x->1 (1 - sqrtx)/(5 - sqrt(1+24x))

If we divide the numerator and denominator by x - 1, we get:
lim (x-->1) (1 - √x)/[5 - √(1 + 24x)]
= lim (x-->1) [(1 - √x)/(x - 1)]/{[5 - √(1 + 24x)]/(x - 1)}
= [lim (x-->1) (1 - √x)/(x - 1)]/{lim (x-->1) [5 - √(1 + 24x)]/(x - 1)}.

Then, note that, by the definition of the derivative:
(i) d/dx √x (@ x = a) = lim (x-->a) (√x - √a)/(x - a)
==> 1/(2√a) = lim (x-->a) (√x - √a)/(x - a)
(ii) d/dx √(1 + 24x) (@ x = a) = lim (x-->a) [√(1 + 24x) - √a]/(x - a)
==> 12/√(1 + 24a) = lim (x-->a) [√(1 + 24x) - √(1 + 24a)]/(x - a).

Letting a = 1 gives:
(a) 1/2 = lim (x-->1) (√x - 1)/(x - 1)
==> lim (x-->1) (1 - √x)/(x - 1) = -1/2
(b) 12/5 = lim (x-->1) [√(1 + 24x) - 5]/(x - 1)
==> lim (x-->1) [5 - √(1 + 24x)]/(x - 1) = -12/5.

Therefore:
[lim (x-->1) (1 - √x)/(x - 1)]/{lim (x-->1) [5 - √(1 + 24x)]/(x - 1)}
= (-1/2)/(-12/5)
= 5/24.

I hope this helps!