Difficulty: Introductory | Prerequisites: Basic algebra, familiarity with function notation.
Limits are the gateway concept of calculus. Before you can understand derivatives or integrals, you need a solid grasp of what it means for a function to approach a value as its input gets close to some number. This topic covers how to read limits off a graph (including one-sided limits), how piecewise-defined functions behave at their breakpoints, and what vertical asymptotes look like algebraically. If you are comfortable with function notation and can sketch basic graphs, you are ready for this material.
A limit describes the value a function approaches as x gets close to some target, regardless of whether the function actually equals that value at the target. One-sided limits look at the approach from the left or right only, and the two-sided limit exists only when both sides agree. Piecewise functions often have interesting limit behaviour at their breakpoints, and vertical asymptotes occur where a function's output grows without bound.
Limit of a function
The value that f(x) approaches as x approaches a particular number c. Written lim (x→c) f(x) = L. The function does not need to equal L at x = c; it only needs to get arbitrarily close to L as x gets close to c.
Think of it as: where the graph is heading, not where it actually lands.
One-sided limit (left-hand limit)
The value f(x) approaches as x approaches c from values less than c (from the left). Written lim (x→c⁻) f(x).
In simple terms, this means: cover the right side of the graph and look only at what happens as you walk toward c from the left.
One-sided limit (right-hand limit)
The value f(x) approaches as x approaches c from values greater than c (from the right). Written lim (x→c⁺) f(x).
In simple terms, this means: cover the left side and look only at the approach from the right.
Two-sided limit
The ordinary limit lim (x→c) f(x). It exists only when the left-hand and right-hand limits both exist and are equal.
Think of it as: both sides of the graph must agree on the same y-value for the two-sided limit to exist.
Piecewise function
A function defined by different formulas on different intervals of its domain. Each "piece" applies over a specified range of x-values.
In simple terms, this means: the rule for computing f(x) changes depending on which interval x falls in.
DNE (Does Not Exist)
Used when a limit has no single finite value. Common reasons: the left and right limits disagree, or the function oscillates, or the output grows without bound.
Think of it as: there is no one number the function is settling toward.
Vertical asymptote
A vertical line x = a where the function's output grows without bound (toward +∞ or −∞) as x approaches a. On the graph, the curve shoots up or down near that x-value and never crosses the line.
In simple terms, this means: the function blows up near x = a because something in the denominator is hitting zero.
Open circle (on a graph)
A hollow dot indicating that the function is not defined at that point, or that the function's value there differs from the limit. The graph approaches this point but does not include it.
Think of it as: "the graph gets here, but the function's actual value is somewhere else (or nowhere)."
Reading the y-value the curve approaches, not the dot
Trace the curve toward the target x-value from both sides.
The limit is the y-value the curve is heading toward, even if there is a hole (open circle) there.
If the function has a filled dot at a different y-value, that is f(c), not the limit.
When a limit equals the function value
If the curve passes smoothly through a filled dot at x = c, then lim (x→c) f(x) = f(c). No conflict.
When a limit exists but differs from f(c)
Example: the curve approaches y = 3 as x→1, but there is an open circle at (1, 3) and a filled dot at (1, 1). The limit is 3, but f(1) = 1.
When a limit does not exist (from a graph)
The curve approaches different y-values from the left and right (a jump). Left-hand limit ≠ right-hand limit, so the two-sided limit is DNE.
The curve shoots off toward ±∞ (a vertical asymptote).
Left-hand limit, lim (x→c⁻) f(x)
Approach x = c using only x-values that are slightly less than c.
On the graph, move along the curve from the left toward the vertical line x = c and read the y-value you are approaching.
Right-hand limit, lim (x→c⁺) f(x)
Approach x = c using only x-values that are slightly greater than c.
On the graph, move along the curve from the right toward x = c.
Connection between one-sided and two-sided limits
lim (x→c) f(x) exists and equals L if and only if both lim (x→c⁻) f(x) = L and lim (x→c⁺) f(x) = L.
If the two one-sided limits are different numbers, the two-sided limit is DNE.
Notation shorthand
The superscript minus (⁻) means "from the left" (values below c).
The superscript plus (⁺) means "from the right" (values above c).
Evaluating limits at a breakpoint
A breakpoint is an x-value where the formula changes. To find the limit at a breakpoint, compute the one-sided limits using the formulas on each side.
Example: f(x) = { x + 4 if x ≤ 0, 4 if x > 0 }. At x = 0, the left-hand limit uses x + 4, giving 0 + 4 = 4. The right-hand limit uses the constant 4, also giving 4. Both sides agree, so lim (x→0) f(x) = 4.
When a piecewise limit does not exist
Example: f(x) = { 12 if x < −5, −x + 7 if −5 ≤ x < 2, 3 if x = 2, 7 if x > 2 }.
At x = −5: left-hand limit = 12 (from the constant piece), right-hand limit = −(−5) + 7 = 12. Both agree, so lim (x→−5) f(x) = 12.
At x = 2: left-hand limit = −2 + 7 = 5 (using the −x + 7 piece), right-hand limit = 7 (from the constant piece). They disagree, so lim (x→2) f(x) = DNE.
The function value at a breakpoint may differ from the limit
In the example above, f(2) = 3 (a separately defined value), while neither one-sided limit equals 3. The limit and the function value are independent questions.
Graphing piecewise functions to check your work
Sketch each piece over its interval. Use filled dots for values the function actually takes, open circles for values approached but not attained.
A jump (gap between pieces) signals that the two-sided limit at that breakpoint is DNE.
Students often think the limit equals the function value. It does not have to. The limit is about where the curve is heading; f(c) is the actual output at c. These can differ, and one can exist without the other.
Students often assume that if a graph has a hole (open circle), the limit does not exist there. The opposite is usually true: the limit exists at the hole, because the curve still approaches a definite y-value. The hole just means the function is not defined there (or is defined elsewhere).
Students sometimes read one-sided limits backwards. The minus superscript (⁻) means approaching from the left (smaller x-values), not that the limit itself is negative.
Students often forget to check both sides at a piecewise breakpoint. Computing only the left-hand or only the right-hand piece and reporting that as the two-sided limit misses the point of the question.
⚠️ Graph-based limit questions appear on nearly every Calculus I exam. You will be given a graph and asked for several limits, including at least one that is DNE.
⚠️ Piecewise function limits are a staple of WeBWorK assignments and exams. Expect to compute one-sided limits at each breakpoint and determine whether the two-sided limit exists.
⚠️ Know the difference between f(c) and lim (x→c) f(x). Exam questions frequently ask for both, and the answers can be different.
⚠️ "DNE" is a valid answer. Do not try to force a numerical answer when the left and right limits disagree or when the function blows up.
True or False: If lim (x→2⁻) f(x) = 5 and lim (x→2⁺) f(x) = 5, then lim (x→2) f(x) = 5. (True)
True or False: If f(3) = 7, then lim (x→3) f(x) = 7. (False, the limit depends on nearby values, not the value at the point.)
Fill in the blank: The notation lim (x→c⁻) means approaching c from the ____. (left)
True or False: A function can have a limit at a point where it is not defined. (True)
Fill in the blank: For f(x) = { x + 4 if x ≤ 0, 4 if x > 0 }, lim (x→0) f(x) = ____. (4)
Q: For f(x) = { 12 if x < −5, −x + 7 if −5 ≤ x < 2, 3 if x = 2, 7 if x > 2 }, find lim (x→−5) f(x).
A: The left-hand limit (using the piece 12) is 12. The right-hand limit (using −x + 7 at x = −5) is −(−5) + 7 = 12. Both agree, so the limit is 12.
Q: Using the same piecewise function, find lim (x→2) f(x).
A: The left-hand limit (using −x + 7 at x = 2) is 5. The right-hand limit (using the constant 7) is 7. They disagree, so the limit is DNE.
Q: If a graph shows the curve approaching y = 3 from both sides as x→1, but there is an open circle at (1, 3) and a filled dot at (1, −4), what is lim (x→1) f(x)? What is f(1)?
A: lim (x→1) f(x) = 3 (where the curve heads). f(1) = −4 (the filled dot).
Q: For f(x) = { x + 4 if x ≤ 0, 4 if x > 0 }, do the left-hand and right-hand limits at x = 0 agree?
A: Left-hand limit: 0 + 4 = 4. Right-hand limit: 4. Yes, they agree. lim (x→0) f(x) = 4.
Q: A graph has the curve going to y = 2 from the left and y = 5 from the right as x→1. What are the one-sided limits, and what is the two-sided limit?
A: lim (x→1⁻) f(x) = 2, lim (x→1⁺) f(x) = 5, lim (x→1) f(x) = DNE.
Limits feed directly into the definition of the derivative. The derivative of f at a point is itself a limit: lim (h→0) [f(x+h) − f(x)] / h. If you cannot evaluate limits confidently, the derivative definition will be opaque.
Continuity (Section 1.6 in most Calculus I courses) is defined in terms of limits: f is continuous at c when lim (x→c) f(x) = f(c). Every continuity question is a limit question in disguise.
Vertical asymptotes connect to the behaviour of rational functions, which you will revisit when studying curve sketching and integration of rational functions later in the course.
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