Evaluate the limit, if it exists.
lim x->0 ((9-x)-(9/(x^2-x)))
To find the limit as x approaches 0, we evaluate the expression directly by substituting 0 for x:
lim x->0 ((9-x)-(9/(x^2-x)))
= (9-0) - (9/(0^2-0))
= 9 - (9/0)
Since division by 0 is undefined, the limit does not exist.
To evaluate the limit as x approaches 0 of ((9-x)-(9/(x^2-x))), we can try to simplify the expression and see if we can find a value for the limit.
Let's start by simplifying the expression:
((9-x)-(9/(x^2-x)))
= (9-x) - (9/(x(x-1))) [factoring out the common denominator]
Now, let's find the common denominator for the expression:
x(x-1)
= x^2 - x
Now, substitute the common denominator back into the expression:
(9-x) - (9/(x^2 - x))
= (9-x) - (9/(x^2 - x))(x^2 - x)/(x^2 - x) [multiplying numerator and denominator by (x^2 - x)]
= (9-x) - (9(x^2 - x)/(x^2 - x))
= (9-x) - 9x^2 + 9x / (x^2 - x) [expanding the numerator]
Simplifying further, we get:
= (9-x) - 9x^2 + 9x / x(x-1)
Now, let's factor out x from the numerator:
= (9 - x - 9x^2 + 9x) / x(x-1)
= (9x^2 - 8x + 9) / x(x-1)
Now, as x approaches 0, we can substitute x = 0 in the expression:
(9(0)^2 - 8(0) + 9) / 0(0-1)
= (0 - 0 + 9) / 0(0-1)
= 9 / 0(0-1)
= 9 / 0
Since we have a denominator of 0, the limit does not exist.
To evaluate the limit as x approaches 0 of the given expression, we can simplify the expression and plug in the value of x.
Let's start by simplifying the expression:
(9 - x) - (9 / (x^2 - x))
First, let's simplify the denominator by factoring out an x:
x^2 - x = x(x - 1)
Now, let's rewrite the expression:
(9 - x) - (9 / (x(x - 1)))
Next, let's find the least common denominator (LCD) for the expression:
The LCD is x(x - 1) because it is the smallest common multiple of both x and (x - 1).
Now, let's rewrite the expression in terms of the LCD:
[(9 - x) * (x(x - 1))] / (x(x - 1)) - (9 / (x(x - 1)))
Next, let's simplify the expression further:
[(9x - x^2 - 9)] / (x(x - 1))
We can factor out a negative 1 from the numerator:
[-(x^2 - 9x + 9)] / (x(x - 1))
Now, let's check if there are any common factors between the numerator and the denominator:
There are no common factors between the numerator and the denominator.
Finally, we can plug in x = 0 into the simplified expression to evaluate the limit:
lim x->0 ((9-x)-(9/(x^2-x))) = lim x->0 (-(x^2 - 9x + 9)) / (x(x - 1))
Substituting x = 0:
= -(0^2 - 9(0) + 9) / (0(0 - 1))
= -(9) / (0)
= undefined.
Therefore, the limit as x approaches 0 does not exist for the given expression.