For f(x) = x² when x < 2 and f(x) = x + 1 when x ≥ 2, what is lim_{x→2} f(x)?
AThe limit does not exist — the left-hand limit is 4 but the right-hand limit is 3
B4 — substitute into the x < 2 piece
C3 — substitute into the x ≥ 2 piece
D2 — the input value is 2, so the limit is 2
The two-sided limit exists only when both one-sided limits agree. From the left: lim_{x→2⁻} x² = 4. From the right: lim_{x→2⁺} (x+1) = 3. Since 4 ≠ 3, the two-sided limit does not exist — this is a jump discontinuity. Options B and C each use only one side; option D confuses the input value with the limit value. The two-sided limit requires consensus from both directions.
Question 2 Multiple Choice
A student evaluates a function and finds lim_{x→5⁺} g(x) = 7. She concludes that lim_{x→5} g(x) = 7. What error has she made?
AShe computed only the right-hand limit; the two-sided limit also requires lim_{x→5⁻} g(x) = 7
BNothing — if the right-hand limit exists and equals 7, the two-sided limit is automatically 7
CShe should have computed lim_{x→5⁻} instead, since the two-sided limit is defined by the left approach
DThe two-sided limit requires the function value g(5) to also equal 7
The two-sided limit requires both one-sided limits to exist AND agree. Knowing the right-hand limit gives only half the picture. The function could have lim_{x→5⁻} g(x) = 3, in which case the two-sided limit would not exist despite the right-hand limit being defined. The student has made the classic error of treating one-sided existence as sufficient for two-sided existence.
Question 3 True / False
If lim_{x→a⁻} f(x) = 5 and lim_{x→a⁺} f(x) = 5, then lim_{x→a} f(x) = 5.
TTrue
FFalse
Answer: True
This is the precise theorem: the two-sided limit equals L if and only if both the left-hand limit and the right-hand limit equal L. When both one-sided limits exist and agree on the same value, the two-sided limit exists and equals that value. Note that f(a) itself could be undefined, different from 5, or equal to 5 — the limit only depends on the behavior near a, not at a.
Question 4 True / False
In the notation lim_{x→a⁻} f(x), the minus superscript means x is approaching a negative value.
TTrue
FFalse
Answer: False
The superscript '−' means 'from the left' — that is, x approaches a through values strictly less than a (e.g., lim_{x→5⁻} means x takes values like 4.9, 4.99, 4.999...). It has nothing to do with a being negative. You can write lim_{x→(−3)⁻} for a left-hand limit at a = −3, or lim_{x→100⁻} for a left-hand limit at a positive number. The sign indicates direction of approach, not the sign of a.
Question 5 Short Answer
Explain in your own words why a function can have both one-sided limits exist at a point, yet the two-sided limit fails to exist.
Think about your answer, then reveal below.
Model answer: The two-sided limit requires the function to be approaching a single agreed-upon value from both directions simultaneously. If the left-hand limit and right-hand limit converge to different values, then as x approaches a from different sides, f(x) is heading toward two different targets — there is no single value that f(x) gets arbitrarily close to. The two-sided limit captures the idea of approach from all directions at once.
The sign function sgn(x) = |x|/x is the classic example: it approaches −1 from the left and +1 from the right at x = 0. Both one-sided limits exist (well-defined targets), but the two-sided limit fails because the two sides disagree. The function is 'doing something different' on each side of 0. A piecewise function with different formulas on each side often produces this situation.