Limits from Graphs
A limit describes the value that a function approaches as its input gets closer and closer to a particular number. Rather than asking 'what is the function equal to at this point?', we ask 'what value is the function heading toward?' Graphs are one of the most powerful tools for visualising and estimating limits because they let us see the behaviour of a function at a glance, including situations where the function itself may be undefined at the point of interest. When reading limits from a graph, you trace along the curve from both the left side and the right side toward the target x-value and observe where the y-values appear to be going. If both sides of the curve approach the same height, the limit exists and equals that height. If the two sides approach different heights — or if the graph shoots off to infinity — the limit does not exist at that point. This idea is captured by the concept of one-sided limits: the left-hand limit and the right-hand limit. Graphical limit estimation is especially useful for piecewise functions, functions with holes (removable discontinuities), and functions with jump or infinite discontinuities. Even when you cannot evaluate a limit algebraically — perhaps because the algebra is complex or the function is given only as a picture — the graph gives you a reliable estimate and a strong conceptual understanding of what limits really mean.
- Identify the target x-value
- Trace the left-hand limit
- Trace the right-hand limit
- Compare both sides
- Distinguish the limit from the function value
- Recognise infinite limits and vertical asymptotes
Identify the target x-value
Locate the x-value (called 'a') at which you want to find the limit. Draw a vertical dashed line at x = a. This is the input value you are approaching, not necessarily a value at which the function is defined.
Trace the left-hand limit
Starting from the left of x = a, follow the graph moving rightward toward x = a. The y-value your finger approaches is the left-hand limit, written lim_{x→a⁻} f(x). Focus on where the graph is heading, not on any isolated point or open circle directly at x = a.
Trace the right-hand limit
Now start from the right of x = a and follow the graph moving leftward toward x = a. The y-value approached from this direction is the right-hand limit, written lim_{x→a⁺} f(x). Again, ignore the actual plotted point at x = a itself.
Compare both sides
If lim_{x→a⁻} f(x) = lim_{x→a⁺} f(x) = L, then the two-sided limit exists: lim_{x→a} f(x) = L. If the two one-sided limits are different, the two-sided limit does not exist (DNE). A jump discontinuity on the graph is a visual signal of DNE.
Distinguish the limit from the function value
A filled circle (closed dot) on a graph at (a, b) means f(a) = b. An open circle (hollow dot) means the function is not defined there. The limit can equal L even if f(a) is undefined, equals something else, or has an open circle — the limit is about approach, not arrival. For example, if there is an open circle at (2, 3) but the graph approaches y = 3 from both sides, then lim_{x→2} f(x) = 3 even though f(2) is undefined.
Recognise infinite limits and vertical asymptotes
If the graph rises or falls without bound as x → a, the limit is ±∞. We write lim_{x→a} f(x) = +∞ or −∞. Technically this means the limit does not exist as a real number, but we use the infinity notation to describe the behaviour precisely. A vertical asymptote at x = a is the visual signature of this behaviour.
The Key Equation
- x \to a — x approaches a (dimensionless): x gets arbitrarily close to the value a but does not necessarily equal a.
- L — Limit value (dimensionless): The real number that f(x) approaches; read off the y-axis on the graph.
- x \to a^{-} — Left-hand approach (dimensionless): x approaches a from values smaller than a (from the left on the graph).
- x \to a^{+} — Right-hand approach (dimensionless): x approaches a from values larger than a (from the right on the graph).
Worked Example
Problem. A piecewise function is graphed. For x < 2, the graph is a line that approaches the point (2, 5) (open circle). For x > 2, the graph is a curve that also approaches (2, 5) (open circle). At x = 2 exactly, there is a filled dot at (2, 1). Using the graph, determine: (a) lim_{x→2⁻} f(x), (b) lim_{x→2⁺} f(x), (c) lim_{x→2} f(x), and (d) f(2).
Answer. The two-sided limit is 5; the function value is 1. These are not equal, illustrating a removable discontinuity.
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
Cauchy was the first mathematician to give a rigorous definition of the limit, forming the bedrock of modern calculus. He replaced the informal notion of 'infinitely small quantities' used by Newton and Leibniz with the precise epsilon-delta framework published in his 1821 textbook Cours d'analyse. This formal definition, which states that a limit L is the value approached by f(x) to within any prescribed error ε when x is within δ of a, is the standard definition used in every calculus course today. Cauchy was extraordinarily prolific, publishing 789 papers during his lifetime — second only to Euler in sheer volume of mathematical output. He even had a habit of submitting multiple papers per week to the French Academy of Sciences, prompting the Academy to impose a page limit on individual submissions.
Related Concepts
- Continuity of Functions — Continuity is defined using limits: a function is continuous at x = a when the limit equals the function value. Understanding graphical limits directly prepares you to identify continuous and discontinuous behaviour.
- Evaluating Limits Algebraically — Once you can read limits from a graph, the next step is computing them using algebra — factoring, rationalising, and limit laws — to get exact values rather than estimates.
- One-Sided Limits — Left-hand and right-hand limits are the building blocks of the two-sided limit seen on every graph. A deeper study of one-sided limits clarifies jump discontinuities and piecewise-defined functions.
- Definition of the Derivative — The derivative is itself a limit — specifically the limit of a difference quotient. Graphical limit intuition is essential for understanding the slope of a tangent line as the limiting case of a secant line.