Finding Limits Numerically and Graphically
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 2.2: “The Limit of a Function” |
| Book URL | https://openstax.org/details/books/calculus-volume-1 |
Freely available and openly licensed.
Key idea
The limit of $f(x)$ as $x$ approaches $a$ asks a question about nearby behavior, not about the value at $a$. You can observe that behavior in two ways before you know any algebra: by building a table of function values with $x$ close to $a$, or by reading a graph and asking what height the curve seems to be heading toward.
These numerical and graphical approaches will not prove a limit is a specific number, but they build genuine intuition, expose hidden behavior (such as oscillation or blowup), and help you check algebraic answers for sanity. They are also exactly how scientists and engineers estimate limits when closed-form algebra is impossible.
Prerequisite Check
Before this lesson, make sure you can do all of the following:
Quick Reference
Informal definition. $\lim_{x \to a} f(x) = L$ means: $f(x)$ can be made as close to $L$ as desired by taking $x$ sufficiently close to $a$ (but not equal to $a$).
Numerical approach. Build a table: choose $x$-values approaching $a$ from both sides and compute $f(x)$ for each. If the values cluster toward a single number $L$ from both sides, the limit is likely $L$.
Graphical approach. Trace the curve from the left toward $(a, ?)$ and from the right toward $(a, ?)$. If both traces head to the same height $L$, the limit is $L$.
Key Concepts
1. Estimating a Limit with a Table
Build a table with $x$ approaching $a$ from the left (values below $a$) and from the right (values above $a$). Compute $f(x)$ at each.
Example 1. Estimate $\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ numerically.
Note: the function is undefined at $x = 2$ (denominator is zero). The limit still asks what value $f(x)$ approaches as $x$ gets close to $2$.
| $x$ | $f(x) = \dfrac{x^2-4}{x-2}$ |
|---|---|
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 1.999 | 3.999 |
| 2.001 | 4.001 |
| 2.01 | 4.01 |
| 2.1 | 4.1 |
Both sides of the table converge to 4. The limit appears to be 4. (Algebraically: $\frac{x^2-4}{x-2} = x+2$ for $x \neq 2$, and $2 + 2 = 4$.)
Example 2. Estimate $\displaystyle\lim_{x \to 0} \frac{\sin x}{x}$ numerically ($x$ in radians).
| $x$ | $\sin(x)/x$ |
|---|---|
| $-0.1$ | 0.99833 |
| $-0.01$ | 0.99998 |
| $-0.001$ | 0.9999998 |
| $0.001$ | 0.9999998 |
| $0.01$ | 0.99998 |
| $0.1$ | 0.99833 |
Both sides converge to 1. The limit is 1. (This is proved rigorously using the squeeze theorem.)
2. What a Table Cannot Tell You
Numerical estimates can be misleading. Two important cautions:
Caution 1: Don’t stop too soon. The function $f(x) = \sin(\pi/x)$ oscillates between $-1$ and $1$ infinitely often as $x \to 0$. A table with $x = 0.1, 0.01, 0.001$ might happen to give values near 0 (depending which $x$-values you chose), suggesting the limit is 0 -- but the limit does not exist because the function oscillates without settling.
Caution 2: Round-off error. Very close to $a$, floating-point arithmetic can introduce errors. Evaluating $\frac{\sqrt{x+1} - 1}{x}$ at $x = 10^{-15}$ on a typical calculator gives 0 (due to round-off), not the true value near $1/2$.
The table approach builds intuition and provides evidence, but it cannot prove a limit. Algebraic or theoretical arguments are required for certainty.
3. Reading Limits from Graphs
On a graph of $y = f(x)$:
- $\lim_{x \to a^-} f(x)$: the height the curve approaches as $x$ moves right toward $a$ (from the left).
- $\lim_{x \to a^+} f(x)$: the height the curve approaches as $x$ moves left toward $a$ (from the right).
- $\lim_{x \to a} f(x)$: exists if and only if both one-sided limits are equal.
Graphical vocabulary:
- An open circle at $(a, L)$ means $f(a) \neq L$ (or $f(a)$ is undefined) but the limit equals $L$ as $x \to a$.
- A filled circle at $(a, L)$ means $f(a) = L$.
- A gap or jump between the left and right traces indicates the limit does not exist.
- A vertical asymptote at $x = a$ means at least one one-sided limit is $\pm\infty$.
Example 3. Suppose the graph of $f$ shows:
- The curve approaching height 3 as $x$ moves toward 2 from the left.
- The curve approaching height 3 as $x$ moves toward 2 from the right.
- An open circle at $(2, 3)$ and a filled circle at $(2, 1)$.
Then: $\lim_{x \to 2^-} f(x) = 3$, $\lim_{x \to 2^+} f(x) = 3$, $\lim_{x \to 2} f(x) = 3$, but $f(2) = 1$.
The limit exists and equals 3 even though $f(2) = 1$. The limit only depends on nearby values, not the value at the point.
Example 4. Suppose the graph of $g$ shows:
- The curve approaching height 5 from the left of $x = 4$.
- The curve approaching height 2 from the right of $x = 4$.
Then: $\lim_{x \to 4^-} g(x) = 5$, $\lim_{x \to 4^+} g(x) = 2$. Since $5 \neq 2$, $\lim_{x \to 4} g(x)$ does not exist. The graph has a jump discontinuity at $x = 4$.
4. Function Value vs. Limit: Three Possible Relationships
| Situation | Limit | $f(a)$ | Continuous at $a$? |
|---|---|---|---|
| Curve passes through $(a, L)$ | $L$ | $L$ | Yes |
| Open circle at $(a, L)$; filled circle at $(a, c)$, $c \neq L$ | $L$ | $c$ | No (removable discontinuity) |
| Left and right limits differ | DNE | anything | No (jump discontinuity) |
| Vertical asymptote at $a$ | DNE ($\pm\infty$) | undefined | No (infinite discontinuity) |
Common Errors
| Error | Example | Correction |
|---|---|---|
| Evaluating $f(a)$ and calling it the limit | “Since $f(2) = 1$, the limit is 1” when there is an open circle at $(2,3)$ | Read the limit from the graph’s approach, not the dot’s position |
| Reading the limit only from one side | Seeing the left approach is 5 and writing $\lim = 5$ | Check both sides; if they differ, the two-sided limit does not exist |
| Trusting a table with only one column | Using only $x > a$ values | Always approach from both sides |
| Confusing a removable discontinuity with no limit | “There’s a hole, so there’s no limit” | The limit exists at a removable discontinuity; only the function value is missing |
Leveled Practice
Level 1 -- Reading from a Table
Problem 1. The following table gives values of a function $f$:
| $x$ | 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 |
|---|---|---|---|---|---|---|
| $f(x)$ | 2.71 | 2.97 | 2.997 | 3.003 | 3.03 | 3.31 |
(a) Estimate $\lim_{x \to 1} f(x)$. (b) What is $f(1)$ based on this table?
Show answer
(a) Values approach 3 from both sides. The limit appears to be $3$.
(b) The table does not tell you $f(1)$; the limit is not $f(1)$. Additional information about $f(1)$ is needed.
Problem 2. Build a table to estimate $\displaystyle\lim_{x \to 0} (1 + x)^{1/x}$ using $x = \pm 0.1, \pm 0.01, \pm 0.001$. Describe what you observe.
Show answer
| $x$ | $(1+x)^{1/x}$ |
|---|---|
| $-0.1$ | $\approx 2.868$ |
| $-0.01$ | $\approx 2.732$ |
| $-0.001$ | $\approx 2.7196$ |
| $0.001$ | $\approx 2.7183$ |
| $0.01$ | $\approx 2.7048$ |
| $0.1$ | $\approx 2.5937$ |
Both sides approach approximately $2.718...$, which is $e$. The limit is $e$.
Level 2 -- Reading from a Graph Description
Problem 3. A graph of $h$ shows:
- For $x < 3$: the curve follows $y = x^2 - 5$ and approaches $(3, 4)$ from the left, with an open circle at $(3, 4)$.
- For $x > 3$: the curve follows $y = 2x - 2$ and approaches $(3, 4)$ from the right.
- A filled circle at $(3, -1)$.
Find $\lim_{x \to 3^-} h(x)$, $\lim_{x \to 3^+} h(x)$, $\lim_{x \to 3} h(x)$, and $h(3)$.
Show answer
$\lim_{x \to 3^-} h(x) = 4$, $\lim_{x \to 3^+} h(x) = 4$, $\lim_{x \to 3} h(x) = 4$, $h(3) = -1$.
The limit exists (both sides agree at 4) but $h(3) \neq \lim_{x \to 3} h(x)$; this is a removable discontinuity.
Level 3 -- Analysis
Problem 4. Explain why estimating $\lim_{x \to 0} \sin(\pi/x)$ from a table might be misleading. What behavior does this function actually exhibit near $x = 0$?
Show answer
As $x \to 0$, $\pi/x \to \pm\infty$, so $\sin(\pi/x)$ oscillates between $-1$ and $1$ infinitely often. Any finite table must miss most of these oscillations and may report values near 0 or 1 depending on which $x$-values happened to be chosen. The limit does not exist because the function never settles to a single value.
Problem 5. A student claims that the limit of $f$ as $x \to 3$ is 7 because “I plugged in $x = 3$ and got $f(3) = 7$.” Describe one situation where this argument is correct and one where it is wrong.
Show answer
Correct: If $f$ is continuous at $x = 3$ (e.g., a polynomial), then $\lim_{x \to 3} f(x) = f(3) = 7$ by the direct substitution property. Here, the student’s reasoning works and gives the right answer.
Wrong: If $f$ is a piecewise function where the formula for $x = 3$ assigns value 7, but the left and right limits equal (say) 5, then $\lim_{x \to 3} f(x) = 5 \neq 7$. The limit is determined by nearby values, not by $f(3)$.
Common Misconceptions
plugging in $x = a$ always gives the limit. The tempting reasoning is that $\lim_{x \to a} f(x) = f(a)$ in every case. This works for polynomials and other continuous functions, but it fails for piecewise functions, functions with holes, and functions not defined at $a$. The limit asks where $f(x)$ is heading as $x$ gets close to $a$, not what value is assigned at $x = a$ itself. A table built from $x$-values near $a$ will reveal the correct limiting behavior even when $f(a)$ does not agree.
a limit is a value the function can never actually reach. Some students picture a limit as a forbidden boundary the function always stays short of. This is not correct. Many functions reach their limit: $\lim_{x \to 2} (3x - 1) = 5$, and the function equals 5 at $x = 2$. The barrier picture is accurate only for cases like $\lim_{x \to 0} \sin(x)/x$, where $x = 0$ is not in the domain. A graph or table distinguishes the two cases immediately.
Mastery Checklist
You have mastered this skill when you can do all of the following without referring to notes:
Mental Model
A limit is about the journey, not the destination.
When you ask $\lim_{x \to a} f(x)$, you are asking: “Where is $f(x)$ heading as $x$ gets close to $a$?” The value $f(a)$ -- if it exists at all -- is the destination, but the limit is the journey. The journey can head to a place the curve never actually reaches (open circle), or it can reach a place and then jump away ($f(a) \neq L$).
A table shows the journey numerically: values of $x$ getting closer and closer to $a$, and the corresponding $f(x)$ values forming a pattern. A graph shows the journey visually: two arrows approaching $a$ from opposite sides. The limit exists when both arrows point to the same height.
Connections
Within Calculus I (MATH161)
- Limit intuition: The graphical and numerical approaches here build the intuition that formal definitions (limit laws, epsilon-delta) make precise.
- Continuity: A function is continuous at $a$ when $f(a) = \lim_{x \to a} f(x)$; the graphical criterion is that there is no open circle, jump, or vertical asymptote at $a$.
- Infinite limits: When the table values grow without bound or the graph shows a vertical asymptote, you are observing an infinite limit, covered in the next lesson.