Please help me find the limit:
lim x--> 2
((1/3)-1/sqrt(x))/)x-9)
1/3 - 1/sqrt 2 = (sqrt 2 -3)/3 sqrt 2
divide that by 2-9 = -7
(sqrt 2 -3)/-21
I recognize this as
Lim ( 1/3 - 1/√x)/(x-9) , as x -->9
But for yours x --->2
Most teachers would tell you that the first step in any limit question is to actually sub in the restricted value into your expression.
If you get 0/0, then you have some work to do
If you get an actual real number, that is your answer and you are done!
so if x = 2 we get
(1/3 - 1/√3)/(-7) which is a real number
so that's it! We are done
All you have to do is simplify the answer
= (√3 - 3)/(-21√3)
= (3 - √3)/(21√3) *(√3/√3)
= (√3-1)/21
go with Damon's
I really don't know how my 1/√2 became 1/√3
To find the limit as x approaches 2 of the given expression, ((1/3) - 1/sqrt(x))/(x - 9), we can use algebraic manipulation and direct substitution. Here's how you can solve it step by step:
Step 1: Simplify the expression.
First, let's simplify the denominator. Since we are taking the limit as x approaches 2, we evaluate the denominator for x = 2:
x - 9 = 2 - 9 = -7
Now, simplify the numerator. To do this, we need to find a common denominator between 1/3 and 1/sqrt(x). Since 1/sqrt(x) is the same as sqrt(x)/x, the common denominator is 3x.
(1/3) - 1/sqrt(x) = (1/3) - sqrt(x)/x
Now we can rewrite the expression as:
[((1/3) * x - sqrt(x))/(3x)] * [-1/7]
= [-x/3 - sqrt(x)] / (21x)
Step 2: Substitute the value of x.
Now, let's substitute x = 2 into the simplified expression:
[-2/3 - sqrt(2)]/(21*2)
Step 3: Evaluate the expression.
Calculating the expression:
[-2/3 - sqrt(2)]/(21*2) ≈ -0.465
Therefore, the limit as x approaches 2 of ((1/3) - 1/sqrt(x))/(x - 9) is approximately -0.465.