- Topic
- Continuity
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Continuity practice question 6
The graph of a function is shown. Which statement about at is true?
Cover the graph to the right of and follow the left piece towards : what height does it approach? Then do the same from the right.
The filled point is the value , which a limit never uses. The limit exists only when the two one-sided limits agree.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Read each one-sided limit from the piece on that side, ignoring what happens at x = 2 itself. The limit exists only if they agree, and continuity needs that limit to equal f(2).
Working
From the left the bowl rises towards the open circle at (2, 3).
Just right of 2 the line is close to the filled point at (2, 1).
The one-sided limits disagree: the graph jumps at x = 2.
The value is defined, but with no limit for it to equal, f is not continuous at 2. It is continuous from the right only.
Answer
The answer is the left-hand limit is 3, the right-hand limit is 1, the limit does not exist, and f is not continuous at 2.
Check
At x = 1.9 the left piece is at about 2.9 and at x = 2.1 the right piece is at 1.1, closing in on 3 and 1.
Where this goes wrong
Reading the limit off the filled point. f(2) = 1 is the value at 2; the limit is about the heights the graph approaches from both sides, and here the two sides head to different heights.
Leans on
One-sided limits, from Limits
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