A graphical limit estimate uses the graph of a function near an approach point \(a\).
Instead of starting with algebra, we inspect what \(f(x)\) appears to approach as \(x\) gets closer to \(a\).
1. Two-sided graphical limit
A two-sided limit exists when the graph approaches the same height from both sides:
\[
\lim_{x\to a}f(x)=L
\]
This means:
\[
\lim_{x\to a^-}f(x)=L
\quad\text{and}\quad
\lim_{x\to a^+}f(x)=L
\]
2. Left-hand and right-hand limits
The left-hand limit looks only at values with \(x\lt a\):
\[
\lim_{x\to a^-}f(x)
\]
The right-hand limit looks only at values with \(x\gt a\):
\[
\lim_{x\to a^+}f(x)
\]
If the left-hand and right-hand limits are different, then the two-sided limit does not exist.
3. Reading a limit from a graph
To estimate a limit graphically:
- Find the approach value \(a\) on the \(x\)-axis.
- Trace the graph toward \(a\) from the left.
- Trace the graph toward \(a\) from the right.
- Compare the two heights.
- If both sides approach the same height, that height is the estimated limit.
4. Removable holes
A function may be undefined at \(x=a\), but the limit can still exist. For example:
\[
f(x)=\frac{x^2-4}{x-2}
\]
At \(x=2\), the expression is undefined. But for \(x\ne2\):
\[
\frac{x^2-4}{x-2}
=
\frac{(x-2)(x+2)}{x-2}
=
x+2
\]
So the graph approaches:
\[
\lim_{x\to2}\frac{x^2-4}{x-2}=4
\]
This is a removable discontinuity. The graph has a hole at \(x=2\), but the limit is still \(4\).
5. Jump discontinuities
A jump occurs when the left and right sides approach different values:
\[
\lim_{x\to a^-}f(x)\ne\lim_{x\to a^+}f(x)
\]
In this case:
\[
\lim_{x\to a}f(x)\text{ does not exist}
\]
6. Infinite behavior
Sometimes the graph grows upward or downward without bound near \(a\). For example:
\[
\lim_{x\to0^+}\frac1x=+\infty
\]
and:
\[
\lim_{x\to0^-}\frac1x=-\infty
\]
Since the two sides behave differently, the two-sided limit does not exist.
7. Numerical tables
A numerical table supports the graph by listing values close to \(a\).
For example, if \(x\) approaches \(2\) from the left:
\[
x=1.9,\ 1.99,\ 1.999,\ \ldots
\]
If the corresponding values of \(f(x)\) get closer to the same number, that number is a likely limit.
8. Confidence in a graphical estimate
| Confidence |
Meaning |
What to check |
| High |
Left and right values are stable and agree closely. |
The graph and table both support the same estimate. |
| Medium |
The values are fairly stable, but the evidence is not perfect. |
Zoom closer and inspect more rows. |
| Low |
The values do not clearly settle. |
The limit may not exist, or more algebra may be needed. |
9. Trace line
A trace line is a vertical line at a selected \(x\)-value. It helps connect the graph to a numerical value:
\[
x=t,\qquad f(t)
\]
Moving the trace line closer to \(a\) helps show whether the function values approach a stable height.
10. Common mistakes
- Do not confuse \(f(a)\) with \(\lim_{x\to a}f(x)\). They can be different.
- A hole in the graph does not automatically mean the limit does not exist.
- A jump means the two-sided limit does not exist.
- Always compare the left-hand and right-hand behavior.
- Zooming in can reveal behavior hidden in a wide graph window.
- Oscillating functions may not settle to one value near the approach point.
