lim ((x^2-64)/(x+8))
x->-8
In calculating the limit of a rational fraction, if the denominator does not evaluate to zero, the limit is found be simply substituting the limit of x for x.
If the denominator evaluates to zero, and so does the numerator, we can see if there is a common factor. If there is, we can cancel the common factor and proceed as another rational fraction (or a polynomial).
As x->8, the denominator evaluates to zero, so we check the numerator. The numerator also evaluates to zero.
We note that the numerator factors into (x+8)(x-8), which has a common factor of (x+8) with the denominator.
Cancelling the common factor leaves us with (x-8), which is a polynomial that can be evaluated by simple substitution:
Lim x->-8 (x-8) = -16.
Therefore
Lim x->-8 (x^2-64)/(x+8) = -16
To find the limit of the given expression as x approaches -8, we can substitute -8 into the expression and evaluate the result. However, if direct substitution yields an undefined or indeterminate form, we'll need to simplify the expression further before substituting the value of x.
Let's evaluate the given expression by substituting x = -8:
((x^2-64)/(x+8)) = ((-8)^2-64)/(-8+8) = (64-64)/0
Here, we obtain an indeterminate form of 0/0, which means we cannot directly evaluate the limit using substitution.
To simplify the expression further, we can factor the numerator using the difference of squares formula.
x^2 - 64 = (x-8)(x+8)
Let's rewrite the expression with the factored numerator:
((x-8)(x+8))/(x+8)
Now, we can cancel out the common factor of (x+8) in the numerator and denominator:
(x-8)/(1)
Finally, we can substitute x = -8 into the simplified expression:
((-8)-8)/(1) = (-16)/1 = -16
Therefore, the limit of the given expression as x approaches -8 is -16.