- Topic
- Limits
- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Limits practice question 11
Evaluate the limit.
The standard result is . It only applies when the argument of sine matches the denominator exactly.
Rewrite the expression so that appears underneath the sine, and put the leftover constant out in front.
A decimal or a fraction.
Theory UBC Math 100 §2.1.1CLP-1 §1.4Key results
Worked solution
Strategy
Reshape the quotient until it contains sin(u)/u with matching arguments, then pull every constant outside the limit.
Working
Multiply and divide by 3 so the denominator matches the argument of sine.
As x → 0 the argument 3x → 0 as well, so the standard limit applies directly.
The constant factor passes straight through the limit.
Answer
The answer is 3/5, or 0.6.
Check
At x = 0.01 the quotient is sin(0.03)/0.05 ≈ 0.59991, just under 0.6.
Where this goes wrong
Answering 1 by treating any sin(ax)/(bx) as the standard limit. The result is 1 only when the two arguments agree; otherwise the answer is a/b.
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