The equation is x^2+6x+9/x^4-81
Limit is x➡️-3
I factored it out and I got 1/36 would that be right
The numerator would factor out to be (x+3)(x+3) then the denominator would be (x+3)(x-3)(x+3)(x-3) this would cancel out the (x+3) which would leave 1/x^2+6x+9 the plugging in -3 I got 1/36 I don't know if that is right
If you canceled out the x+3 terms, that leaves the x-3 terms, so your quotient is
1/(x-3)^2
But your answer is indeed 1/36
X^2 +6x+9/x^4-81
I put the answer as 1/36 it's not the answer??
It told me to use the identity x^4-a^4=(x+a)(x-a)(x^2+a^2) I used it u still get the same answer....
To find the limit of the given expression, you correctly factored the numerator and denominator.
The numerator, x^2 + 6x + 9, can be further simplified as (x + 3)(x + 3) or (x + 3)^2.
The denominator, x^4 - 81, can be factorized as (x^2 + 9)(x^2 - 9), and further simplified as (x + 3)(x - 3)(x^2 + 3)(x^2 - 3).
Now, cancelling out the common factor (x + 3) from the numerator and denominator, we are left with:
[(x + 3)(x + 3)] / [(x + 3)(x - 3)(x^2 + 3)(x^2 - 3)]
As x approaches -3, we can substitute -3 into the simplified expression to find the limit:
[( -3 + 3)( -3 + 3)] / [( -3 + 3)( -3 - 3)( -3^2 + 3)( -3^2 - 3)]
Simplifying further:
[0] / [0]
At this point, we have an indeterminate form of 0/0. To determine the limit, we can apply L'Hôpital's rule or further simplify the expression.
Using L'Hôpital's rule, you would differentiate the numerator and denominator separately and then evaluate the limit again. However, in this case, we can simplify the expression by canceling out one of the common factors, (x - 3), carefully to avoid dividing by zero.
After canceling (x - 3), the simplified expression becomes:
1 / [(x + 3)(x^2 + 3)(x^2 - 3)]
Now, substituting -3 into the simplified expression:
1 / [(-3 + 3)(-3^2 + 3)(-3^2 - 3)]
Simplifying further:
1 / [0]
In this case, we still have an indeterminate form of 1/0. Therefore, the limit as x approaches -3 does not exist.
Hence, your answer of 1/36 is not correct. The limit does not converge to any specific value.