The function gg is given by g(x)=7x−26x−5g(x)=7x−26x−5. The function hh is given by h(x)=3x+142x+1h(x)=3x+142x+1. If ff is a function that satisfies g(x)≤f(x)≤h(x)g(x)≤f(x)≤h(x) for 0
To find limx→2f(x), we need to determine the behavior of f(x) as x approaches 2
To find limx→2f(x), we need to determine the behavior of f(x) as x approaches 2
To find limx→2f(x), we need to determine the behavior of f(x) as x approaches 2.
Given that g(x) ≤ f(x) ≤ h(x), we can rewrite the inequality as:
7x – 2 ≤ f(x) ≤ 3x + 14
Now, let’s evaluate this inequality at x = 2:
7(2) – 26(2) – 5 ≤ f(2) ≤ 3(2) + 14
-12 ≤ f(2) ≤ 20
Therefore, we have -12 ≤ f(2) ≤ 20.
Since this is an inequality, we cannot determine the exact value of f(2). However, we can determine the limit as x approaches 2.
To find the limit, we need to consider the behavior of f(x) as x approaches 2 from both the left and the right.
As x approaches 2 from the left (x < 2), the values of f(x) approach -12 according to the inequality. So, the left-hand limit would be lim{x→2-}f(x) = -12. As x approaches 2 from the right (x > 2), the values of f(x) approach 20 according to the inequality. So, the right-hand limit would be lim{x→2+}f(x) = 20.
Since the left-hand limit and the right-hand limit are not equal, the limit as x approaches 2 for f(x) does not exist (DNE).
Therefore, lim{x→2}f(x) is undefined.
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