Left-Hand and Right-Hand Limits
Mental model
If the left-hand limit and right-hand limit do not match exactly, the limit does not exist.
For absolute value functions, the rule changes at zero. You must check the left path and right path separately to see if they meet at the same point.
Explanation
Some functions split into two different rules at a point. You cannot just substitute the value. You must evaluate the limit as you approach from the left, and separately from the right. If these two numbers are different, the paths do not meet, and the overall limit does not exist.
Derivation
Evaluate the limit of |x|/x as x approaches 0
- \lim_{x \to 0^-} \frac{|x|}{x} — modulus-left — Start by finding the left-hand limit as x approaches 0.
- \lim_{x \to 0^-} \frac{-x}{x} — modulus-left — For x less than 0, the absolute value of x is defined as -x.
- \lim_{x \to 0^-} (-1) — cancel-common-factors — Cancel x from the top and bottom.
- -1 — limit-of-constant — The limit of a constant is the constant itself.
- \lim_{x \to 0^+} \frac{|x|}{x} — modulus-right — Now find the right-hand limit as x approaches 0.
- \lim_{x \to 0^+} \frac{x}{x} — modulus-right — For x greater than 0, the absolute value of x is defined as x.
- \lim_{x \to 0^+} (1) — cancel-common-factors — Cancel x from the top and bottom.
- 1 — limit-of-constant — The limit of a constant is the constant itself.
- \text{Limit does not exist} — lhl-rhl-compare — The left-hand limit (-1) does not equal the right-hand limit (1).
Common mistakes
- Forcing limit existence: Students think every math question must have a number answer, so they write 0 or 1. 'Does not exist' is a valid math answer when paths do not meet.
Recall questions
- Explain when a limit does not exist in your own words, as if to a classmate.
- How does |x| open when x approaches zero from the left side?
- What does the minus sign mean in x → 2⁻?
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