题目

MAT135H5_F25_ALL SECTIONS 2.4 Preparation Check

多重下拉选择题

Consider the function 𝑓 ( 𝑥 ) = { 𝑥 2 + 1 𝑖 𝑓 𝑥 < 2 3 𝑖 𝑓 𝑥 = 2 7 − 𝑥 𝑖 𝑓 𝑥 > 2 . We aim to find out if 𝑓 ( 𝑥 ) has a discontinuity at 𝑥 = 2 , and if so, of what type. In order to do that, first find the following information: 𝑓 ( 2 ) = [ Select ] 5 3 2 7 lim 𝑥 ⟶ 2 − 𝑓 ( 𝑥 ) = [ Select ] 7 3 2 5 lim 𝑥 ⟶ 2 + 𝑓 ( 𝑥 ) = [ Select ] 5 7 2 3 Is 𝑓 ( 𝑥 ) continuous or discontinuous at 𝑥 = 2 ? [ Select ] discontinuous continuous If 𝑓 ( 𝑥 ) is discontinuous at 𝑥 = 2 , what type of discontinuity is it? [ Select ] f is continuous An infinite discontinuity A removable discontinuity A jump discontinuity

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思路分析

We are given a piecewise function f(x): - f(x) = x^2 + 1 if x < 2 - f(x) = 3 if x = 2 - f(x) = 7 − x if x > 2 and several prompts to fill in for x = 2. First, restating the information: - f(2) = [Select] corresponds to the actual value of the function at x = 2, which is 3 according to the definition. - lim_{x → 2⁻} f(x) = [Select] is the left-hand limit as x approaches 2 from below. Since x < 2 uses x^2 + 1, this tends to 2^2 + 1 = 5, so the left-hand limit is 5. - lim_{x → 2⁺} f(x) = [Select] is the right-hand limit as x approaches 2 fr......Login to view full explanation

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类似问题

Suppose that with a certain phone company, an international long distance phone call from Canada to Brazil costs $0.90 for the first minute (up to and including 60 seconds), plus $0.50 for each additional minute or part of a minute. Note: "Part of a minute" means that if a new minute is started even just by one second, a full minute is charged. For example, a 5 min 1 sec phone call costs the same as a 5 min 50 sec phone call and the same as a 6 min 0 sec phone call. Suppose 𝐶 ( 𝑡 ) is the function that gives the cost of making a 𝑡 minute long phone call. On a piece of paper, sketch a graph showing 𝐶 ( 𝑡 ) (with 𝐶 on the 𝑦 -axis and 𝑡 on the 𝑥 -axis). Then use your graph to evaluate each of the following: (Write DNE for undefined.) 𝐶 ( 2.5 ) = [ Select ] DNE 1.9 1.4 2.4 2.9 𝐶 ( 4 ) = [ Select ] 3.4 1.9 2.9 2.4 DNE lim 𝑥 → 3.1 𝐶 ( 𝑡 ) = [ Select ] 1.9 1.4 2.9 2.4 DNE lim 𝑥 → 4 − 𝐶 ( 𝑡 ) = [ Select ] 2.4 1.4 2.9 1.9 DNE lim 𝑥 → 4 + 𝐶 ( 𝑡 ) = [ Select ] DNE 2.9 2.4 3.4 1.9 lim 𝑥 → 4 𝐶 ( 𝑡 ) = [ Select ] 1.9 2.9 DNE 3.4 2.4

Consider this graph of the function 𝑓 ( 𝑥 ) . Which of the following statements are true and which are false? lim 𝑥 → 3 − 𝑓 ( 𝑥 ) = lim 𝑥 → 3 + 𝑓 ( 𝑥 ) [ Select ] False True lim 𝑥 → 1 𝑓 ( 𝑥 ) = 𝑓 ( 1 ) [ Select ] True False 𝑓 ( 𝑥 ) has a vertical asymptote at 𝑥 = 4 . [ Select ] False True 𝑓 ( 𝑥 ) has a vertical asymptote at 𝑥 = 6 . [ Select ] False True lim 𝑥 → 4 − 𝑓 ( 𝑥 ) = lim 𝑥 → 4 + 𝑓 ( 𝑥 ) [ Select ] False True lim 𝑥 → 4 𝑓 ( 𝑥 ) = ∞ [ Select ] True False lim 𝑥 → 6 𝑓 ( 𝑥 ) = ∞ [ Select ] False True The limit lim 𝑥 → 4 𝑓 ( 𝑥 ) exists, but lim 𝑥 → 6 𝑓 ( 𝑥 ) does not exist. [ Select ] False True

Consider this graph of the function 𝑓 ( 𝑥 ) . Which of the following statements are true and which are false? lim 𝑥 → 3 − 𝑓 ( 𝑥 ) = lim 𝑥 → 3 + 𝑓 ( 𝑥 ) [ Select ] False True lim 𝑥 → 1 𝑓 ( 𝑥 ) = 𝑓 ( 1 ) [ Select ] False True 𝑓 ( 𝑥 ) has a vertical asymptote at 𝑥 = 4 . [ Select ] True False 𝑓 ( 𝑥 ) has a vertical asymptote at 𝑥 = 6 . [ Select ] False True lim 𝑥 → 4 − 𝑓 ( 𝑥 ) = lim 𝑥 → 4 + 𝑓 ( 𝑥 ) [ Select ] True False lim 𝑥 → 4 𝑓 ( 𝑥 ) = ∞ [ Select ] False True lim 𝑥 → 6 𝑓 ( 𝑥 ) = ∞ False The limit lim 𝑥 → 4 𝑓 ( 𝑥 ) exists, but lim 𝑥 → 6 𝑓 ( 𝑥 ) does not exist. [ Select ] True False

MTH1010_09_10_3

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