Final answer:
To evaluate the limit, factor both the numerator and the denominator, cancel common factors and substitute x=4 into the reduced expression, giving a final answer of 3/4.
Step-by-step explanation:
The question asks to evaluate the limit lim (x² - 2x - 8) / (x²-16) as x approaches 4. This is a standard calculus problem involving limits. Initially, substitution of x=4 is impossible due to a zero denominator. To solve, we need to simplify the fraction.
Both the numerator and denominator are quadratic expressions, so we look for factors. The numerator factors to (x-4)(x+2) and the denominator to (x-4)(x+4). Canceling the common factor (x-4) gives:
(x+2)/(x+4). Now, substituting x=4, we get:
6/8 which simplifies to 3/4.
The given expression is a limit expression. In order to find the value of the limit, we can use the fact that if a polynomial has the same degree for both numerator and denominator, then the limit can be found by dividing the coefficients of the highest degree terms. In this case, both the numerator and denominator have a degree of 2.
So, the limit can be found by dividing the coefficient of the highest degree term in the numerator (which is 1) by the coefficient of the highest degree term in the denominator (which is also 1). Therefore, the limit is 1.
lim (x² - 2x - 8) / (x²-16) = 1 when x→4.