- Topic
- Limits
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Limits practice question 3
Suppose and . Evaluate the limit below.
The limit laws let you take the limit of each piece separately, as long as the denominator’s limit is not zero. Is it?
The denominator tends to , which is not 0. In the numerator, replace by 4 and by , minding the sign of .
A whole number.
Theory UBC Math 100 §2.1.1CLP-1 §1.4Key results
Worked solution
Strategy
Limits of sums, products and quotients are the sums, products and quotients of the limits, provided the denominator does not tend to zero. Check that, then substitute the two given limits.
Working
The quotient law applies because the denominator has a non-zero limit.
Power and constant-multiple laws on the numerator.
Quotient of the two limits.
Answer
The answer is 11.
Check
With f(x) = 4 + x and g(x) = −2 + x near a = 0, the quotient at x = 0.001 is about 10.99, close to 11.
Where this goes wrong
Sign slips with g: −3g(x) tends to −3 × (−2) = +6, not −6. Write the substitution out before simplifying.
Leans on
Substituting into an expression, from Functions, graphs and inverses
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