Let f(x)=−(x−1)21.Select the correct description of the one-sided limits of f at x=1.Choose 1 answer:(A)limx→1+f(x)=+∞ and limx→1−f(x)=+∞(B)limx→1+f(x)=+∞ and limx→1−f(x)=−∞(c)limx→1+f(x)=−∞ and limx→1−f(x)=+∞(D)limx→1+f(x)=−∞ and limx→1−f(x)=−∞
Q. Let f(x)=−(x−1)21.Select the correct description of the one-sided limits of f at x=1.Choose 1 answer:(A)limx→1+f(x)=+∞ and limx→1−f(x)=+∞(B)limx→1+f(x)=+∞ and limx→1−f(x)=−∞(c)limx→1+f(x)=−∞ and limx→1−f(x)=+∞(D)limx→1+f(x)=−∞ and limx→1−f(x)=−∞
Understand function behavior: Analyze the function f(x)=−(x−1)21 to understand its behavior near x=1. The function is a rational function with a negative sign in front and a squared denominator. This means that as x approaches 1, the denominator approaches 0, which will cause the value of the function to approach infinity. However, the negative sign will affect the direction of the infinity (positive or negative).
Calculate right-hand limit: Calculate the right-hand limit as x approaches 1 from the right (x→1+).As x gets closer to 1 from the right, (x−1) becomes a small positive number. Squaring a small positive number gives a small positive number. Dividing −1 by a small positive number gives a large negative number. Therefore, the right-hand limit is negative infinity.limx→1+f(x)=−∞
Calculate left-hand limit: Calculate the left-hand limit as x approaches 1 from the left (x→1−).As x gets closer to 1 from the left, (x−1) becomes a small negative number. Squaring a small negative number still gives a small positive number (since the square of any real number is non-negative). Dividing −1 by a small positive number again gives a large negative number. Therefore, the left-hand limit is also negative infinity.limx→1−f(x)=−∞
Describe one-sided limits: Combine the results from Step 2 and Step 3 to describe the one-sided limits of f at x=1. Both the right-hand and left-hand limits of f as x approaches 1 are negative infinity. Therefore, the correct description of the one-sided limits of f at x=1 is: limx→1+f(x)=−∞ and limx→1−f(x)=−∞
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