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Questions and Answers
The open dot on the graph at x=3 indicates that f(x) is continuous at x=3.
The open dot on the graph at x=3 indicates that f(x) is continuous at x=3.
False
The function h(x) is discontinuous at x=0 because the limit as x approaches 0 from the left is different from the limit as x approaches 0 from the right.
The function h(x) is discontinuous at x=0 because the limit as x approaches 0 from the left is different from the limit as x approaches 0 from the right.
True
The function f(x) exists at x=3 because there is a closed dot on its graph at that point.
The function f(x) exists at x=3 because there is a closed dot on its graph at that point.
False
The function h(x) is continuous at x=0 because lim┬(x→0)〖h(x)〗 exists.
The function h(x) is continuous at x=0 because lim┬(x→0)〖h(x)〗 exists.
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The function f(x) is continuous at x=3 because f(3) exists.
The function f(x) is continuous at x=3 because f(3) exists.
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Study Notes
Continuity of Functions
- 𝑓(𝑥) is discontinuous at 𝑥=3 because the open dot on the graph indicates that 𝑓(𝑥) does not exist at 𝑥=3, violating the first condition of continuity.
Continuity of ℎ(𝑥)
- ℎ(0) exists and is equal to 1.
- The left-hand limit of ℎ(𝑥) as 𝑥 approaches 0 from the left is -1, denoted by lim┬(𝑥→0^− )〖ℎ(𝑥)〗=−1.
- The right-hand limit of ℎ(𝑥) as 𝑥 approaches 0 from the right is 1, denoted by lim┬(𝑥→0^+ )〖ℎ(𝑥)〗=1.
- Since the left-hand and right-hand limits are not equal, the limit of ℎ(𝑥) as 𝑥 approaches 0 does not exist, violating the second condition of continuity.
- Therefore, ℎ(𝑥) is discontinuous at 𝑥=0.
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Description
Test your understanding of continuity and discontinuity of functions by analyzing graphs and determining points of discontinuity. Explore properties such as existence of limits and the conditions for a function to be continuous.
