- Topic
- Continuity
- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Continuity practice question 10
Find the value of that makes continuous at .
Continuity at needs three things to agree: the left-hand limit, the right-hand limit, and .
Compute each side separately. The left piece gives a number; the right piece gives an expression in . Set them equal.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
A piecewise function is continuous at a join point when the two pieces approach the same value there. Evaluate both one-sided limits and force them to agree.
Working
Approaching from the left uses the first piece.
Approaching from the right uses the second piece, which also defines f(1).
Continuity requires the two sides to meet.
Solve. With a = 6 both sides give 3, and f(1) = 3 as well.
Answer
The answer is a = 6.
Check
Either side of 1, at x = 0.999 and 1.001, the pieces give 2.998 and about 3.012, both close to 3.
Where this goes wrong
Setting a(1)² = 3 and answering a = 3, which drops the −3 from the second piece. Substitute into the whole expression, not just the term containing a.
Leans on
One-sided limits, from Limits
Try a similar question
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