Lim x to 3 (-x^2)/((x^2)- 6x + 9)
substitution i get -9/0
factoring i get same answer
would that mean the answer is infinity?
if not how could i prove that it isn't infinity
Often if you do not trust your answer, try numbers for example
say x = 3.1
-9.61 / (9.61 - 18.6 + 9)
= - 9.61 / .01 = -961
next try x = 3.01
but you get the idea. I think your answer is correct. The result is -9/0 which is undefined or infinite.
since x^2-6x+9 = (x-3)^2 the denominator -> 0 while the numerator does not. So, you are correct that the limit is undefined.
To determine the limit of the function, you can try to simplify the expression by factoring and canceling common factors. However, in this case, substituting the value directly into the expression leads to an indeterminate form of -9/0, which does not provide a clear answer.
To determine whether the limit exists and find its value, you can employ algebraic techniques. In this case, let's first factor the denominator, which is a quadratic expression: (x^2) - 6x + 9 = (x - 3)^2.
Now, the expression becomes:
(-x^2) / ((x - 3)^2).
Since the expression is in the form of a fraction, with a denominator having a square term, we can simplify it further by canceling the common factor:
(-x^2) / ((x - 3) * (x - 3)) = -x^2 / (x - 3).
Now, we can evaluate the limit by substituting the value of x = 3 into the simplified expression:
lim(x→3) (-x^2) / (x - 3).
By substitution, we get:
(-3^2) / (3 - 3) = -9 / 0.
At this point, we still have an indeterminate form of -9/0, and cannot conclude the limit value from this evaluation alone.
To prove that the limit is not infinity, we need to attempt an alternative approach. One possible method is to use the concept of one-sided limits.
Evaluate the left-sided limit by approaching x = 3 from values less than 3:
lim(x→3-) (-x^2)/(x - 3).
Substituting x = 2 into the expression, we get:
(-2^2) / (2 - 3) = -4 / (-1) = 4.
Similarly, evaluate the right-sided limit by approaching x = 3 from values greater than 3:
lim(x→3+) (-x^2)/(x - 3).
Substituting x = 4 into the expression, we get:
(-4^2) / (4 - 3) = -16 / 1 = -16.
Since the left-sided and right-sided limits are not equal (4 != -16), the limit of the function as x approaches 3 does not exist. Therefore, the function does not tend towards infinity.