Let f(x) = (x² − 4)/(x − 2). What is lim_{x→2} f(x)?
AUndefined, because f(2) is undefined (division by zero)
B0, because the numerator equals 0 when x = 2
C4, because the numerator factors as (x−2)(x+2), which simplifies to x+2, and x+2 → 4 as x → 2
DThe limit does not exist because x cannot equal 2
Factor the numerator: x² − 4 = (x−2)(x+2). For x ≠ 2, the (x−2) terms cancel and f(x) = x+2. As x approaches 2 (but never reaches it), x+2 approaches 4. The limit is 4 even though f(2) is undefined — this is exactly the point of limits: they describe approaching behavior, independent of the function's value (or lack thereof) at the target point. Options A and D confuse 'undefined at the point' with 'no limit exists.'
Question 2 Multiple Choice
Define g(x) = x² for all x ≠ 3, and g(3) = 100. What is lim_{x→3} g(x)?
A100, because that is the function's actual value at x = 3
B9, because as x approaches 3, x² approaches 9 — the limit reflects the surrounding behavior, not the isolated value at x = 3
CThe limit does not exist because g has a discontinuity at x = 3
D3, because limits equal the input value as x approaches that input
The limit as x → 3 depends on what g(x) does for x close to (but not equal to) 3. For all x ≠ 3, g(x) = x², and x² → 9 as x → 3. The special value g(3) = 100 is irrelevant to the limit — the limit is about approach, not arrival. This is the core distinction: the limit can exist and differ from f(a), or f(a) may not even be defined. Option A is the classic misconception that conflates the limit with the function's value.
Question 3 True / False
If lim_{x→a} f(x) = L, then f(a) should equal L.
TTrue
FFalse
Answer: False
The limit and the function's value at the point are independent. A limit describes what f(x) approaches as x gets close to a — it explicitly does not depend on f(a). The function might have f(a) = L (making it continuous at a), or f(a) might differ from L (a removable discontinuity), or f(a) might not even be defined (like 0/0 forms). All three situations are compatible with lim_{x→a} f(x) = L existing.
Question 4 True / False
A limit can exist at a point where a function is not defined.
TTrue
FFalse
Answer: True
This is demonstrated by the classic example f(x) = (x² − 1)/(x − 1), which is undefined at x = 1, yet lim_{x→1} f(x) = 2 because f(x) simplifies to x+1 for all x ≠ 1. The limit is about what value f approaches, not what value it takes. This is precisely why limits are essential to calculus: derivatives are defined as limits of difference quotients that are undefined at the exact point in question (0/0 form), yet those limits can still exist.
Question 5 Short Answer
Explain the difference between 'the limit of f(x) as x approaches a' and 'the value f(a).' Why does this distinction matter for calculus?
Think about your answer, then reveal below.
Model answer: The limit lim_{x→a} f(x) = L describes what f(x) approaches as x gets arbitrarily close to a, and is defined entirely by f's behavior near a — not at a. The value f(a) is simply what the function outputs when you plug in x = a exactly. The two can agree (continuity), differ (removable discontinuity), or one can exist without the other. This distinction matters because derivatives are defined as limits of difference quotients that are undefined at the exact point of evaluation — the limit machinery lets us work through that 0/0 form and extract the derivative anyway.
Every major concept in calculus — derivatives, integrals, continuity — rests on limits. The derivative f'(a) = lim_{h→0} [f(a+h) − f(a)]/h requires taking a limit of an expression that is undefined at h = 0. If limits required the expression to be defined at the target point, calculus would be impossible. The decoupling of 'limit' from 'function value' is what makes the whole machinery work.