Calculus
Limits: the intuition before the algebra
A limit asks what value a function approaches as x approaches a point — not what happens at the point. When direct substitution gives 0/0, the expression usually simplifies first.
Why it matters
Limits define both the derivative and the integral. Students who treat limits as a substitution rule struggle later with continuity, asymptotes and rates of change.
Worked example
Evaluate lim(x→3) (x² − 9)/(x − 3)
- 1Try substitution. gives 0/0, an indeterminate form
- 2Factor the numerator. (x − 3)(x + 3)/(x − 3)
- 3Cancel the common factor. x + 3, for x ≠ 3
- 4Substitute now. 3 + 3
Answer: 6
Common mistakes
- Concluding the limit does not exist as soon as substitution gives 0/0.
- Cancelling without noting the restriction x ≠ 3.
- Ignoring one-sided behaviour when the two sides disagree.
How to check your answer
- Evaluate at values very close on each side — the results should converge on your answer.
- A limit exists only when the left and right limits agree.
Related topics
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