A function f is defined everywhere except at x = 2. As x approaches 2, f(x) approaches 7. Which statement is correct?
AThe limit cannot exist because f(2) is undefined.
Blim(x→2) f(x) = 7, and f(2) being undefined is irrelevant to the limit.
CThe limit is undefined because there is a hole in the graph.
DThe limit only exists if we first define f(2) = 7.
Limits describe the trend of f(x) as x approaches a, not the value at x = a. The function need not be defined at a — only near it. A removable discontinuity (hole) is the classic example where the limit exists but the function value does not.
Question 2 True / False
For any function f, the limit lim(x→a) f(x) generally equals f(a).
TTrue
FFalse
Answer: False
This is the most common misconception about limits. Limits describe the behavior near a, which can differ from the value at a. A function with a jump discontinuity, a hole, or any discontinuity at a will have lim(x→a) f(x) ≠ f(a) — or f(a) may not even be defined. Only continuous functions satisfy lim(x→a) f(x) = f(a).
Question 3 Short Answer
A student says: 'I checked f(1.999) and f(2.001) and both give values near 5, so lim(x→2) f(x) = 5.' What is the logical gap in this reasoning?
Think about your answer, then reveal below.
Model answer: Checking two points near a is not sufficient. The limit requires that f(x) approaches L for ALL x sufficiently close to a, not just two sampled points. The trend must hold everywhere in a neighborhood around a.
The definition of a limit is a universal claim about all x close enough to a — not an existential claim about some x close to a. A function could behave well at 1.999 and 2.001 but oscillate wildly between them, preventing the limit from existing. Numerical evidence can support a conjecture but cannot establish a limit.