Evaluate the limit or state that it does not exist.
lim Root(6+x)-Root(6-x)/x
x->0
i kept getting zero which ever way i tried to solve it.. help plz.
multiply top and bottom by (√(6+x) + √(6-x))/(√(6+x) + √(6-x))
to get
Lim (6+x - 6+x)/[x(√(6+x) + √(6-x)] as x --> 0
= lim 2x/[x(√(6+x) + √(6-x)]
= lim 2/[√(6+x) + √(6-x)] as x --0
= 2/(2√6)
= 1/√6
If you have a calculator handy, here is a nice way to check for limits.
Pick a value of x close to the "approach" value
e.g. x = .0001
then evaluate the expression
that gave me .40825
and 1/√6 = .4082489 , not bad eh?
To evaluate the limit, we can simplify the expression first.
Let's start by rationalizing the numerator. We can do that by multiplying both the numerator and denominator by the conjugate of the numerator, which is √(6+x) + √(6-x):
lim (Root(6+x)-Root(6-x))/x
x->0
= lim ((Root(6+x)-Root(6-x))/x) * ((√(6+x)+√(6-x))/(√(6+x)+√(6-x)))
x->0
Now, we can simplify the numerator using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b):
= lim ((6+x) - (6-x))/ (x(√(6+x) + √(6-x)))
x->0
= lim (2x)/(x(√(6+x) + √(6-x)))
x->0
= lim 2/(√(6+x) + √(6-x))
x->0
Now, let's substitute x = 0 into the expression:
= 2/(√(6+0) + √(6-0))
= 2/(√6 + √6)
= 2/(2√6)
= √6/√6
= 1
So, the limit is equal to 1.