- Topic
- Continuity
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Continuity practice question 28
A student decides whether the function below is continuous at as shown. Find the first line that is wrong.
Check each line against the algebra and against the exact definition it uses. Does each claim follow from what has been shown?
Continuity at a point has three parts. Which one has not been checked?
What is , and does it match the limit?
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Corrected working
Strategy
Compute the limit, read off the value, and compare: continuity needs both to exist and to agree.
Working
Cancel x − 1, then substitute.
The value at 1 is given separately.
The limit and the value disagree, so f is not continuous at 1.
Answer
The answer is f is not continuous at 1, because the limit is 2 and the value is 3.
Check
f(0.99) = 1.99 and f(1.01) = 2.01 sit near 2, while f(1) = 3 sits a whole unit above them.
Where this goes wrong
Stopping once the limit exists. A limit that exists is necessary, but continuity also needs f(1) to equal it.
Leans on
Evaluating limits by cancelling, from Limits
Try a similar question
A function is continuous at , and . Which of the following must be true?Saved in this browser only.