- Topic
- Limits
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Limits practice question 2
The function is defined piecewise below. Which statement about its behaviour at is true?
A limit at 1 is about the values of near 1. Which piece of the formula gives those values on each side?
A limit never looks at the value . Compute what approaches from the left using the piece, and from the right using the piece.
Theory UBC Math 100 §2.1CLP-1 §1.3Key results
Worked solution
Strategy
Evaluate each one-sided limit from the piece that applies on that side. The two-sided limit exists only if they agree; the value f(1) is irrelevant to whether it does.
Working
From the left, x < 1, so the first piece applies.
From the right, x > 1, so the third piece applies.
The one-sided limits disagree. f(1) = 3 does not change that.
Answer
The answer is the left-hand limit is 2, the right-hand limit is 3, and the two-sided limit does not exist.
Check
f(0.99) = 1.9801 and f(1.01) = 2.99, so the values head for 2 from the left and 3 from the right.
Where this goes wrong
Reading the limit off the middle line: f(1) = 3 is the value of the function at 1, and a limit is about the values nearby, not the value there.
Leans on
Evaluating a piecewise function, from Functions, graphs and inverses
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