1.3 Estimating Limit Values from Graphs

1.3 Estimating Limit Values from Graphs

A limit can be estimated without knowing the exact value of a function at the target input: it describes the height that $f(x)$ approaches as $x$ gets close to that input. Graphs show this approach visually; tables show it numerically.

Reading a limit from a graph

To estimate $\displaystyle \lim_{x\to a}f(x)$, focus on the behavior of the curve near $x=a$, not automatically on the point plotted at $x=a$. Imagine sliding along the curve from the left and from the right. If both sides approach the same $y$-value, that common value is the estimated limit.

For example, suppose a graph has a hole in the curve at $(6,4)$, while a filled point appears at $(6,9)$. The nearby curve approaches $4$ from both sides. Therefore,

$$ \lim_{x\to 6}f(x)\approx 4, $$

even though

$$ f(6)=9. $$

The limit records nearby behavior; the function value records the actual defined output at the target input.

A negative target input must be read just as carefully. Suppose the graph contains a filled point at $(-6,3)$, but the curve on either side of $x=-6$ approaches $7$. Then

$$ f(-6)=3 $$

but

$$ \lim_{x\to -6}f(x)\approx 7. $$

The minus sign in $x\to -6$ identifies the horizontal location being approached. It is not part of the resulting $y$-value.

Graph-reading procedure

1. Locate the target input $x=a$, including its sign.
2. Trace the curve toward $x=a$ from the left.
3. Trace the curve toward $x=a$ from the right.
4. Compare the two nearby heights.
5. Report the common height, if one exists, using $\displaystyle \lim_{x\to a}f(x)$ notation.

When a graph gives no two-sided limit

The two-sided limit does not exist when the left-hand and right-hand approaches do not agree. A graph may also fail to produce a finite limit if the function grows without bound near the target or oscillates indefinitely.

Named misconception — “The limit is the filled dot.” A filled dot tells you $f(a)$, but it may have no connection to the limit if the surrounding curve approaches a different height. Conversely, an open circle does not mean the limit is nonexistent; the curve may approach the hole from both sides perfectly consistently.

Connecting representations

The same limit may be represented graphically, numerically, analytically, or verbally:

Representation	Information
Graphical	Both branches approach the height $4$ near $x=2$.
Numerical	Table values approach $4$ from the left and right.
Analytical	$\displaystyle \lim_{x\to 2}f(x)\approx4$.
Verbal	“As $x$ gets close to $2$, $f(x)$ gets close to $4$.”

This cross-checking is Connecting Representations, Skill 2.B: Identify mathematical information from graphical, numerical, analytical, and/or verbal representations. A calculator can zoom, generate tables, or evaluate nearby inputs, but the reasoning remains the same: identify the target input, inspect both directions, and determine whether the outputs approach one common value.

Limit or function value?

Retrieval check: A graph has a filled point at $(-6,3)$, while the curve approaches $8$ from both sides of $x=-6$. State both quantities.

The answer is

$$ f(-6)=3 \qquad\text{and}\qquad \lim_{x\to -6}f(x)\approx8. $$