Difficulty: Introductory | Prerequisites: Section 1.1 (Average and Instantaneous Rate of Change), basic understanding of functions and graphs
Limits are the foundation that the rest of calculus is built on. In the previous section you estimated instantaneous rates of change by shrinking intervals between two points. Limits formalise that shrinking process, giving you a precise way to describe what a function's output is "heading toward" as the input approaches a particular value. You need a solid grasp of limits before moving on to continuity (Section 1.3) and, eventually, derivatives.
A limit describes the value a function approaches as its input gets closer and closer to some number, regardless of whether the function is defined there. If the left-hand and right-hand limits agree, the overall limit exists; if they disagree, the limit does not exist (DNE). When function values grow without bound near a point, you have an infinite limit and a vertical asymptote.
Limit
The value that f(x) approaches as x gets closer and closer to a particular number a. Written as lim (x→a) f(x). In simple terms, it is the number the function is "heading toward" near a point, even if it never arrives there.
Left-hand limit
The value f(x) approaches as x approaches a from values less than a (from the left). Written with a superscript minus: lim (x→a⁻) f(x). Think of it as sneaking up on a from below: 2.9, 2.99, 2.999 approaching 3.
Right-hand limit
The value f(x) approaches as x approaches a from values greater than a (from the right). Written with a superscript plus: lim (x→a⁺) f(x). Think of it as approaching a from above: 3.1, 3.01, 3.001 approaching 3.
One-sided limit
Either a left-hand limit or a right-hand limit. These examine behaviour from only one direction.
Does Not Exist (DNE)
A limit is said to "not exist" when the left-hand and right-hand limits are unequal, or when one (or both) of the one-sided limits fails to settle on a finite value in a way that matches the other side. In simple terms, the function is heading toward two different places depending on which direction you come from.
Infinite limit
A limit where f(x) grows without bound (toward +∞ or −∞) as x approaches a particular value. The overall limit DNE in this case, but we describe the one-sided behaviour using ∞ or −∞.
Vertical asymptote
A vertical line x = a where the function's values increase or decrease without bound as x approaches a. On the graph the curve shoots upward or downward near that line without ever crossing it. Think of it as an invisible wall the function races toward but can never touch.
Open circle (on a graph)
Indicates the function is not defined at that point. The point is not part of the graph.
Closed circle (on a graph)
Indicates the function is defined at that point. The point is included on the graph.
A limit asks: "What value is f(x) approaching as x gets closer and closer to a?" The critical point is that we care about what happens near x = a, not necessarily at x = a.
Two reasons this matters:
The function might not even be defined at x = a (e.g. division by zero).
The function might be defined there but its value might differ from the nearby trend.
The limit captures the trend or behaviour of the function as x approaches a, not the function's value at that exact point.
There are two directions from which x can approach a:
From the left (x → a⁻): x takes values less than a, creeping closer. Example: 2.9, 2.99, 2.999 approaching 3.
From the right (x → a⁺): x takes values greater than a, creeping closer. Example: 3.1, 3.01, 3.001 approaching 3.
The two-sided limit lim (x→a) f(x) exists if and only if both one-sided limits exist and are equal:
lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = L → the limit exists and equals L.
Conversely, if the overall limit equals L, then both one-sided limits also equal L.
If the left-hand and right-hand limits are different, or if either one-sided limit fails to exist, then the overall limit Does Not Exist (DNE).
Consider f(x) = x + 4. As x approaches 0:
x | −0.1 | −0.01 | −0.001 | 0 | 0.001 | 0.01 | 0.1 |
|---|---|---|---|---|---|---|---|
f(x) | 3.9 | 3.99 | 3.999 | ? | 4.001 | 4.01 | 4.1 |
From both sides, f(x) approaches 4. The left-hand and right-hand limits are equal, so the limit exists and equals 4.
Consider the piecewise function f(x) = 2x + 1 when x < 2, and f(x) = 4 − x when x ≥ 2.
At x = 2:
Left-hand limit: using f(x) = 2x + 1, values approach 5 (e.g. f(1.999) = 4.998).
Right-hand limit: using f(x) = 4 − x, values approach 2 (e.g. f(2.001) = 2.001 → heading toward 2).
Since 5 ≠ 2, the limit at x = 2 does not exist. Graphically, tracing from the left the curve approaches 5 (open circle), while from the right it approaches 2 (closed circle). This is called a jump discontinuity.
Sometimes, as x approaches a, the function values do not settle toward a finite number. Instead they grow without bound, either positively or negatively. This is an infinite limit.
This function is undefined at x = 3 (denominator equals zero). Investigating values near x = 3:
From the left (x → 3⁻):
x | 2.9 | 2.99 | 2.999 | 2.9999 |
|---|---|---|---|---|
f(x) | −145 | −1,495 | −14,995 | −149,995 |
The values plummet toward −∞.
From the right (x → 3⁺):
x | 3.1 | 3.01 | 3.001 | 3.0001 |
|---|---|---|---|---|
f(x) | 155 | 1,505 | 15,005 | 150,005 |
The values soar toward +∞.
Because one side heads to −∞ and the other to +∞, the overall limit at x = 3 does not exist.
A function has a vertical asymptote at x = a if its values increase or decrease without bound as x approaches a. On the graph, the curve races upward or downward near that vertical line.
In the example above, x = 3 is a vertical asymptote of f(x) = 5x / (x − 3). From the left the graph drops steeply (f(x) → −∞), and from the right it rises steeply (f(x) → +∞).
How to spot a vertical asymptote: look for values of x that make the denominator zero while the numerator stays non-zero. Those are your candidates.
Consider f(x) = (x² − 9) / (x − 3). The function is undefined at x = 3 because the denominator is zero. Yet the limit at x = 3 still exists.
Why? Because (x² − 9) factors as (x − 3)(x + 3). For every x ≠ 3 the function simplifies to x + 3. So as x approaches 3, f(x) approaches 6 from both sides.
x | 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 |
|---|---|---|---|---|---|---|
f(x) | 5.9 | 5.99 | 5.999 | 6.001 | 6.01 | 6.1 |
Both one-sided limits equal 6, so lim (x→3) f(x) = 6. On the graph you see a straight line y = x + 3 with an open circle (hole) at the point (3, 6). The function is not defined there, but the limit still exists.
This is a key insight: the limit depends on behaviour near the point, not at it.
When given a graph of a function, use the following approach to find limits:
Left-hand limit: Trace the curve from the left side toward the target x-value. Whatever y-value the curve is heading toward is the left-hand limit.
Right-hand limit: Trace the curve from the right side. Whatever y-value the curve is heading toward is the right-hand limit.
Overall limit: If both one-sided limits land at the same y-value, the limit exists and equals that value. If they disagree, the limit DNE.
Things to watch for on graphs:
An open circle means the function is not defined there, but the limit may still exist.
A closed circle shows where the function is defined, which may or may not match the limit.
A curve shooting upward or downward without bound signals an infinite limit and a vertical asymptote.
A jump between two different y-values (one from the left, one from the right) means the limit DNE.
\lim_{x \to a} f(x) = LThe two-sided limit: f(x) approaches L as x approaches a.
\lim_{x \to a^{-}} f(x) = LThe left-hand limit: f(x) approaches L as x approaches a from values less than a.
\lim_{x \to a^{+}} f(x) = LThe right-hand limit: f(x) approaches L as x approaches a from values greater than a.
The existence rule:
lim (x→a) f(x) = L if and only if lim (x→a⁻) f(x) = L and lim (x→a⁺) f(x) = L.
Infinite limit notation:
lim (x→a⁻) f(x) = −∞ means the function decreases without bound from the left.
lim (x→a⁺) f(x) = +∞ means the function increases without bound from the right.
In either case the overall two-sided limit does not exist, but we still describe the one-sided behaviour using ∞ or −∞.
Limits underpin every derivative you will ever compute. Whenever an engineer calculates an instantaneous velocity, a physicist finds the rate of radioactive decay, or an economist measures marginal cost, they are evaluating a limit. Vertical asymptotes appear in real systems too: the intensity of light or sound approaching a point source follows an inverse-square law that blows up at distance zero, which is exactly an infinite limit.
Students often think the limit equals the function's value at that point. It does not. The limit is about the trend near the point, and the function may not even be defined there.
Students confuse "the limit is infinity" with "the limit exists." When a function approaches ∞ or −∞, the limit does not exist in the usual sense. We write ∞ to describe the behaviour, but DNE is still the correct verdict for the two-sided limit.
Students sometimes assume that if a function is defined at x = a, the limit must equal f(a). That is only true for continuous functions, which are covered in Section 1.3.
Students forget to check both sides. A left-hand limit on its own does not determine the overall limit. You must verify the right-hand limit matches before concluding the limit exists.
⚠️ You will almost certainly be asked to evaluate limits from a table of values, a graph, or both. Know the process: check left, check right, compare.
⚠️ Piecewise functions are a favourite exam topic. Expect at least one question where the formula changes at the target point, and you need to evaluate each piece separately.
⚠️ Be ready to identify vertical asymptotes from a formula (denominator = 0, numerator ≠ 0) and to state the one-sided infinite limits.
⚠️ "The limit exists" and "the function is defined" are independent facts. You may be asked for a case where one is true and the other is not.
True or false: If f(a) is undefined, then lim (x→a) f(x) cannot exist. Answer: False. The limit can still exist if f(x) approaches the same value from both sides.
Fill in the blank: The limit lim (x→a) f(x) exists if and only if the ____-hand limit and the ____-hand limit are both equal. Answer: left; right.
True or false: If lim (x→a⁻) f(x) = +∞ and lim (x→a⁺) f(x) = +∞, the overall limit equals +∞. Answer: False. We write lim = +∞ to describe the behaviour, but the limit does not exist in the finite sense. (Some courses accept "the limit is +∞" as a valid answer when both sides agree on +∞; check your instructor's convention.)
True or false: A vertical asymptote occurs where the denominator of a rational function equals zero. Answer: True (provided the numerator is non-zero at that point; if both are zero, further analysis is needed).
Fill in the blank: On a graph, an open circle at a point means the function is ____ at that point. Answer: not defined (or has a hole).
Q: Given f(x) = (x² − 4) / (x − 2), find lim (x→2) f(x).
A: Factor the numerator: (x − 2)(x + 2) / (x − 2) = x + 2 for x ≠ 2. As x → 2, x + 2 → 4. The limit is 4.
Q: Consider g(x) = 3x + 1 for x < 5, and g(x) = 2x for x ≥ 5. Does lim (x→5) g(x) exist?
A: Left-hand limit: 3(5) + 1 = 16. Right-hand limit: 2(5) = 10. Since 16 ≠ 10, the limit does not exist.
Q: For h(x) = 1 / (x − 7), describe the behaviour as x → 7 from each side.
A: From the left (x → 7⁻), the denominator is a small negative number, so h(x) → −∞. From the right (x → 7⁺), the denominator is a small positive number, so h(x) → +∞. The overall limit DNE, and x = 7 is a vertical asymptote.
Q: A graph shows a function approaching y = 3 from the left of x = 4 and y = 3 from the right, but the closed circle is at (4, 1). What is lim (x→4) f(x)?
A: The limit is 3. The function's actual value at x = 4 is 1, but limits depend on the trend near the point, not the value at the point.
Q: If lim (x→a⁻) f(x) = 5 and lim (x→a) f(x) = 5, what is lim (x→a⁺) f(x)?
A: It must be 5. If the two-sided limit exists and equals 5, both one-sided limits equal 5.
Limits lead directly into continuity (Section 1.3): a function is continuous at a point precisely when the limit there exists and equals the function's value. Continuity in turn is the precondition for the derivative (Section 2.x), which is itself defined as a limit of the difference quotient. So every derivative you compute later traces back to exactly this material. Vertical asymptotes reappear when you study rational functions and curve-sketching.
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