- Topic
- Limits
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Limits practice question 1
Evaluate the limit.
What happens to the top and to the bottom as ? Decide whether that settles the limit or only tells you to rewrite first.
Substituting gives , which tells you nothing yet. It is a signal to rewrite, not a final answer.
The numerator is a difference of two squares.
Theory UBC Math 100 §2.1.1CLP-1 §1.4Key results
Worked solution
Strategy
Direct substitution gives 0/0, an indeterminate form. That means the numerator and denominator share a factor. Cancel it, then substitute.
Working
Factor the numerator as a difference of squares. The shared factor is now visible.
Cancel. This is valid because a limit only cares about x near 3, never x equal to 3.
The simplified function is continuous, so substitute.
Answer
The answer is 6.
Check
At x = 3.01 the quotient is 0.0601/0.01 = 6.01, close to 6.
Where this goes wrong
Reading 0/0 as “undefined” and concluding the limit does not exist. An indeterminate form means the algebra is not finished; the function is genuinely undefined at x = 3, but the limit as x approaches 3 still exists.
Leans on
Factoring polynomials, from Functions, graphs and inverses
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