Evaluate lim x→0 √(x+ 2) − √(2) / x. Do not draw a graph or make a table of values.

Answer 1

lim {[√(x+ 2) - √(2)]/x} = lim {[√(x+ 2) - √(2)][[√(x+ 2) + √(2)]/x}}{[√(x+ 2) + √(2)x} = 1/(2√2)

x →0 x→0

Answer 2

first try plugging in x=0 but that leads to 0/0 an indeterminate form

so try l'Hopital's rule. take the derivative of numerator & denominator, separately

to get

(1/2)(x+2)^-1/2 over 1 = 1/2sqr(x+2), now plug in x=0 to get 1/2sqr2

Answer 3

This is the definition of the derivative of the function sqrt(y) at y = 2. There are various ways to solve this, but at this point in the year, you probably know that the derivative is 1/2 x^-1/2 at x = 2. or 1/2*(1/sqrt(2)) = sqrt(2)/4.

Answer 4

I will assume I'm reading this correctly and this falls into the following limit:

lim

x-> 0 sqrt(x+2) - sqrt(2)/x

For the purposes of the solution, it doesn't matter if the x on the right is under the radical sign or not.

Right away we see that sqrt(x+2) doesn't have any issues with x = 0 as an input. So if we were just evaluating that part of the function we could just evaluate the function at x = 0. We can single out this part of the function, because the limit of a sum is the same as the limit of each of the individual parts of the function added together. This is due to the linearity of the limit operation itself. Derivatives and integrals in future problems will also obey this rule.

The second term though poses a big problem. What is the behavior of this function (sqrt(2)/x) at zero?

The simple answer is that the limit does not exist for odd powers of x in the denominator. Approaching from the right it diverges to positive infinity, and from the left it diverges oppositely to negative infinity. This is why 1/x is said to be undefined at zero. So this limit does not exist.

Note that if the x is meant to be under the radical above, this still holds true.