Evaluate a rational limit with a removable hole
The idea
Substitute the approach value to identify the indeterminate form 0/0; it is not itself an answer.
Step by step
- Substitute the approach value to identify the indeterminate form 0/0; it is not itself an answer.
- Factor the numerator and cancel a common factor only where that factor is nonzero.
- Evaluate the simplified expression’s limit and distinguish that limit from the original function’s undefined value.
Example
Find the limit as x approaches 2 of (x2 − 4)/(x − 2).
Answer
4
Factor x2 − 4 as (x − 2)(x + 2). For x ≠ 2, the quotient is x + 2, whose limit is 2 + 2 = 4. The original quotient is undefined at 2.
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Make a table of (x2 − 9)/(x − 3) for values just below and above 3. Explain the limiting value and the hole.
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