Practice question · Put in order
Order the steps of evaluating the limit of as x approaches 2.
- Factor the numerator as (x - 2)(x + 2)
- Report the limit as 4
- Substitute x = 2 into the simplified expression x + 2
- Substitute x = 2 and obtain the indeterminate form 0/0
- Cancel the common factor x - 2, which is legitimate because x is never equal to 2
Hints
- Try substitution first; the algebra is only needed when substitution fails.
- The cancellation must be justified before the second substitution is allowed.
Show the answer
- Substitute x = 2 and obtain the indeterminate form 0/0
- Factor the numerator as (x - 2)(x + 2)
- Cancel the common factor x - 2, which is legitimate because x is never equal to 2
- Substitute x = 2 into the simplified expression x + 2
- Report the limit as 4
Why
Substitution is always the first move, and the 0/0 outcome is what licenses the algebra. The justification in step 3 matters: cancelling x - 2 changes the function AT x = 2, but the limit never looks there, so the two functions have the same limit.
Practise Limits
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