Suppose that h(x)={x2−𝑥+5 if 𝑥<25 if 𝑥=2𝑥3−1 if 𝑥>2Which of the following is equal to 7?I. limx→2−ℎ(𝑥)II. limx→2+ℎ(𝑥)III. lim𝑥→2ℎ(𝑥) Suppose that h(x)= ⎩⎪⎨⎪⎧ x 2 −x+55x 3 −1 if if if x<2x=2x>2 Which of the following is equal to 7?I. x→2−lim h(x)II. x→2+lim h(x)III. x→2lim h(x)
Question
Suppose that h(x)={x2−𝑥+5 if 𝑥<25 if 𝑥=2𝑥3−1 if 𝑥>2Which of the following is equal to 7?I. limx→2−ℎ(𝑥)II. limx→2+ℎ(𝑥)III. lim𝑥→2ℎ(𝑥) Suppose that h(x)= ⎩⎪⎨⎪⎧ x 2 −x+55x 3 −1 if if if x<2x=2x>2 Which of the following is equal to 7?I. x→2−lim h(x)II. x→2+lim h(x)III. x→2lim h(x)
Solution
The function h(x) is defined differently for values of x less than 2, equal to 2, and greater than 2.
I. The limit as x approaches 2 from the left (x→2−) would use the function definition for x<2, which is x^2 - x + 5. Substituting 2 into this equation gives 2^2 - 2 + 5 = 7. So, lim_(x→2−) h(x) = 7.
II. The limit as x approaches 2 from the right (x→2+) would use the function definition for x>2, which is 2x^3 - 1. Substituting 2 into this equation gives 2*(2^3) - 1 = 15. So, lim_(x→2+) h(x) = 15, not 7.
III. The limit as x approaches 2 (x→2) does not exist in this case because the left-hand limit (7) does not equal the right-hand limit (15).
So, only I. lim_(x→2−) h(x) is equal to 7.
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