Math · Calculus I · Worked example
Find a limit at an excluded input
Find the limit of (x² − 4)/(x − 2) as x approaches 2.
Identify the approach point and the excluded input
We are finding the limit as x approaches 2. First check where the original expression is defined: its denominator excludes x = 2.
Try substitution and interpret the result
Both numerator and denominator become zero. We cannot divide zero by zero, so substitution has not answered the question. Here, factoring gives us a way to study nearby values.
Factor the numerator
Use the difference-of-squares identity: a² − b² = (a − b)(a + b). Since 4 is 2², the numerator contains the same factor as the denominator.
Cancel only where the factor is nonzero
At nearby inputs other than 2, division by x − 2 is allowed. Keep x ≠ 2 beside the simplified expression so the missing point is not forgotten.
Take the limit of the simpler expression
As x approaches 2, x + 2 approaches 4 from either side. Because the original quotient agrees with x + 2 near the point, it has the same limit.
Separate the limit from the function value
Our answer is 4. The original function is still undefined at 2. In the tool, compare the left and right limits with the function-value result; those entries answer different questions.
| Question | Conclusion |
|---|---|
| Limit from the left | 4 |
| Limit from the right | 4 |
| Two-sided limit | 4 |
| Original function value at 2 | Undefined |
Result
The limit is 4; the original function is undefined at x = 2.
Your turn
Find the limit of (x² − 9)/(x − 3) as x approaches 3. Is the original function defined at 3?
Show the answer and explanation
The limit is 6, and the original function is undefined at 3.
Factor x² − 9 as (x − 3)(x + 3). For x ≠ 3, the quotient equals x + 3, which approaches 6 from both sides. The original denominator still excludes 3.
Keep exploring
Use the workspace to explore a changed input. Return to the concept if you want to revisit the reasoning.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.