Question

Find the limit.

lim
h → 0 square root (49 + h) − 7 / h


Note: Only 49 +h are under square root.

Answer 1

Use L'Hopital's Rule:

sqrt(49+h) = (49+h)^1/2
= 49^1/2 + 1/2 49^-1/2 * h + (1/2)(-1/2)/2 49^-3/2 h^2 ...
= 7 + h/2*7 + ... (higher powers of h)

so,(sqrt(49+h)-7)/h = 1/14 + ... (powers of h, all being 0)

so, limit as h->0 is 1/14

Note that if y=√x, y' = 1/2√x, so y'(7) = 1/14

Answer 2

Ummm. make that Binomial Theorem, not L'Hopital's Rule