What is a Limit?
Mental model
The limit asks what the function approaches near the point, not what the function is exactly at the point.
For ten years, you have used substitution to evaluate functions. A limit is a new tool to find out where a function is heading, even if it is broken exactly at that point.
Explanation
When you substitute a number into a function and get 0/0, the function is broken at that exact point. It has a hole. But the graph still heads somewhere. A limit finds where it is heading. Keep the limit sign in every step until the exact moment you substitute the number.
Derivation
Evaluate the limit of (x^2 - 4)/(x - 2) as x approaches 2
- \lim_{x \to 2} \frac{x^2 - 4}{x - 2} — identify-indeterminate-form — Set up the limit. Substituting 2 now gives zero divided by zero.
- \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2} — factor-difference-of-squares — Factor the top to find the common term that causes the zero.
- \lim_{x \to 2} (x + 2) — cancel-common-factors — Cancel (x-2). We can do this because inside a limit, x approaches 2 but never exactly equals 2.
- 2 + 2 — substitute-limit — Now it is safe to put x = 2. We drop the limit sign here because we are evaluating.
- 4 — substitute-limit — Add the numbers to find the final limit.
Common mistakes
- Dropping limit sign early: Students often drop the limit sign before taking the limit to do algebra faster on the page. Keep the limit sign in every step until the exact moment you substitute the number.
Recall questions
- Explain what a limit is in your own words, as if to a classmate.
- Why can we cancel (x-2) from the numerator and denominator when evaluating the limit as x approaches 2?
- When do you stop writing the limit sign?