Skill: Evaluate limits from graphs

To evaluate $\displaystyle \lim_{x\to a} f(x)$ from a graph, look at the $y$-values that the graph approaches as $x$ gets close to $a$.

1. Follow the graph from the left of $x=a$ and find the value it approaches:

$$\lim_{x\to a^-}f(x)$$

2. Follow the graph from the right of $x=a$ and find the value it approaches:

$$\lim_{x\to a^+}f(x)$$

3. Compare the two values:
   • If they are equal, that common value is the two-sided limit.
   • If they are different, the limit does not exist.
4. Do not focus only on the actual point at $x=a$. An open circle, a missing point, or a different function value does not automatically change the limit.

Worked example

Suppose a graph shows that:

• As $x$ approaches $2$ from the left, the graph approaches $y=3$.
• As $x$ approaches $2$ from the right, the graph also approaches $y=3$.
• There is an open circle at $(2,3)$.
• The graph has a filled point at $(2,5)$, so $f(2)=5$.

Find

$$\lim_{x\to 2}f(x).$$

Step 1: Check the left-hand behavior.

From the graph,

$$\lim_{x\to 2^-}f(x)=3.$$

Step 2: Check the right-hand behavior.

From the graph,

$$\lim_{x\to 2^+}f(x)=3.$$

Step 3: Compare the one-sided limits.

Both sides approach the same value:

$$3=3.$$

Therefore,

$$\boxed{\lim_{x\to 2}f(x)=3}.$$

The filled point at $(2,5)$ tells us that $f(2)=5$, but it does not affect the limit. The limit describes what $f(x)$ approaches near $x=2$, not necessarily the value at $x=2$ itself.

If the graph approached different values from the two sides, the two-sided limit would be DNE (does not exist). If the graph increased or decreased without bound near $x=a$, the graph would indicate an infinite limit near a vertical asymptote.