1.2 Limits from Tables and Graphs
Notes
You can estimate a limit by choosing inputs closer and closer to $a$ from both sides and watching the outputs.
- If the outputs approach the same number from both sides, that number is the limit.
- On a graph, follow the curve toward $x = a$ from the left and the right. An open circle or a filled dot exactly at $x = a$ does not change the limit.
- If the two sides approach different numbers, the limit does not exist (DNE).
- Example: if $f(1.9) = 3.9$, $f(1.99) = 3.99$, $f(2.01) = 4.01$, $f(2.1) = 4.1$, then $\displaystyle \lim_{x \to 2} f(x) = 4$.
Practice
A table shows $f(0.9) = 2.9$, $f(0.99) = 2.99$, $f(1.01) = 3.01$, $f(1.1) = 3.1$. The best estimate of $\displaystyle \lim_{x \to 1} f(x)$ is:
The outputs from both sides approach 3.
A graph approaches the height 3 from both sides of $x = 1$. There is an open circle at $(1, 3)$ and a filled dot at $(1, 5)$. What is $\displaystyle \lim_{x \to 1} f(x)$?
The limit follows the curve near $x = 1$, which heads toward height 3. The dot at height 5 is only the value $f(1)$.
As $x$ approaches 4 from the left, $f(x)$ approaches 2. As $x$ approaches 4 from the right, $f(x)$ approaches 6. What is $\displaystyle \lim_{x \to 4} f(x)$?
The two sides approach different values, so the two-sided limit does not exist.
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