- Topic
- Limits
- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Limits practice question 23
Evaluate the limit.
The cosine factor has no limit at : it oscillates faster and faster. But it is never large.
Cosine is trapped between and . Multiply that inequality through by , which is never negative.
A single number.
Theory UBC Math 100 §2.1.1CLP-1 §1.4Key results
Worked solution
Strategy
The product rule for limits does not apply, because one factor has no limit. Instead, bound the size of the whole product: the cosine is never more than 1 in size, and x² tends to 0.
Working
Cosine is bounded regardless of how wildly its input behaves.
Multiply through by x². The inequality signs are safe because x² ≥ 0.
Both bounds tend to 0.
By the Squeeze Theorem: the product is trapped between −x² and x², and both tend to 0.
Answer
The answer is 0.
Check
At x = 0.01 the product is about −0.00009, and at x = 0.001 about 0.00000015, shrinking toward 0 however the cosine swings.
Where this goes wrong
Concluding the limit does not exist because cos(5/x) oscillates without settling. Oscillation only prevents a limit when the oscillation keeps its size; here it is being multiplied by something shrinking to zero.
Leans on
Bounded functions and inequalities, from Functions, graphs and inverses
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