if f(x)is
greater or equal to 10x–42 but less than or equal to x2+2x–26
determine limx4f(x) =
What theorem did you use to arrive at your answer?
To determine the limit of a function, we need to evaluate the values of the function as x approaches a specific value. In this case, we need to find the limit of f(x) as x approaches 4.
Given that f(x) is greater than or equal to 10x - 42 and less than or equal to x^2 + 2x - 26, we can use the Squeeze Theorem to determine the limit of f(x) as x approaches 4.
The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all x close to a (except possibly at a), and the limit of both f(x) and h(x) as x approaches a is L, then the limit of g(x) as x approaches a is also L.
In our case, considering the given inequalities:
10x - 42 ≤ f(x) ≤ x^2 + 2x - 26
We can focus on the maximum and minimum values of either side. The maximum value of 10x - 42 occurs when x = 4, and it is 10 * 4 - 42 = 38. The minimum value of x^2 + 2x - 26 occurs at x = 4, and it is 4^2 + 2 * 4 - 26 = 12.
So, we have:
38 ≤ f(x) ≤ 12
As x approaches 4, the limit of f(x) is L, which must satisfy 38 ≤ L ≤ 12. However, this is not possible since 38 is greater than 12. Therefore, the limit lim(x→4)f(x) does not exist.
To summarize, I used the Squeeze Theorem to determine that the limit lim(x→4)f(x) does not exist since the maximum value of 10x - 42 is greater than the minimum value of x^2 + 2x - 26.