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Treating 0/0 as a limit value

Direct substitution is a check, not the only method. An indeterminate quotient needs analysis of nearby values.

Spot the wrong turn

Find the limit of 2x + 5 as x approaches 3.

Incorrect answer: 8

0/0 is not a limit value and is not automatically zero. Canceling a factor can help find nearby behavior without making the original function defined at the excluded point.

Work it through

The correction

11

Direct substitution gives 2(3) + 5.

  1. Try substitution and check whether the expression is defined.
  2. For a 0/0 rational form, factor and simplify for inputs near but not equal to the target.
  3. Evaluate the simplified expression and distinguish the limit from the function’s value at the point.

Try the reasoning in another example

Find the limit of (x² − 4)/(x − 2) as x approaches 2.

  1. Substitution gives 0/0. Factor to (x − 2)(x + 2)/(x − 2).
  2. For x ≠ 2 the expression is x + 2, whose limit is 4.

Answer: 4

Explain why each step preserves the quantity or relationship. If you can compute the answer but cannot explain the step, revisit the model before adding more questions.

The short check below reuses the first example as a retrieval question and adds a second question. It checks this method; it does not establish a grade level.

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1 Find the limit of 2x + 5 as x approaches 3.
2 Find the limit of (x² − 1)/(x − 1) as x approaches 1.