Answer :
The evaluation of the three conditions, we can conclude that the function is continuous at x = 7 because the function is defined at x = 7, the limit as x approaches 7 exists, and the value of the function at x = 7 is equal to the limit.
The given function is f(x) = x^2 - 10x + 21.
To determine if the function is continuous at x = 7, we need to use the definition of continuity. According to the definition, a function is continuous at a point if three conditions are satisfied:
1. The function is defined at that point.
2. The limit of the function as x approaches that point exists.
3. The value of the function at that point is equal to the limit.
Let's evaluate these conditions for x = 7:
1. The function is defined at x = 7 because we can substitute 7 into the function: f(7) = 7^2 - 10(7) + 21 = 49 - 70 + 21 = 0.
2. To determine if the limit as x approaches 7 exists, we need to evaluate the left-hand limit (lim x→7- f(x)) and the right-hand limit (lim x→7+ f(x)). Let's calculate these limits:
a. Left-hand limit (lim x→7- f(x)):
We substitute values slightly less than 7 into the function to evaluate the left-hand limit:
lim x→7- f(x) = lim x→7- (x^2 - 10x + 21)
= (7 - ε)^2 - 10(7 - ε) + 21, where ε is a small positive number.
= 49 - 14ε + ε^2 - 70 + 10ε + 21
= ε^2 - 4ε
As ε approaches 0, the left-hand limit equals 0.
b. Right-hand limit (lim x→7+ f(x)):
We substitute values slightly greater than 7 into the function to evaluate the right-hand limit:
lim x→7+ f(x) = lim x→7+ (x^2 - 10x + 21)
= (7 + ε)^2 - 10(7 + ε) + 21, where ε is a small positive number.
= 49 + 14ε + ε^2 - 70 - 10ε + 21
= ε^2 + 4ε
As ε approaches 0, the right-hand limit equals 0.
Both the left-hand limit and the right-hand limit exist and are equal to 0.
3. The value of the function at x = 7 is f(7) = 0.
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