Question

1.) Lim [√(x + 1) - (2)] / (x - 3)
x -> 3


2.) Lim [ (1/ x + 4) - (1 / 4)] / (x)
x -> 0

Answer 1

Take the ratio of the derivatives of numerator and denominator, and evaluate it at the x value in question.

For both problems, the derivative of the denominator is just 1, so you just have to evaluate the derivative of the numerator.

In the first problem,
Lim [ã(x + 1)-(2)]/(x-3)
x -> 3

= (1/2ã(x + 1)
= 1/4 at x =3

For the second problem
Lim [(1/(x+4)-(1/4)]/ x
x -> 0
= -1(x+4)^2 at x=0
= -1/16