1.1 The Idea of a Limit
Notes
A limit describes the value a function gets close to as the input gets close to some number. We write $\displaystyle \lim_{x \to a} f(x) = L$.
- The limit depends on what happens near $x = a$, not necessarily at $x = a$.
- $f(a)$ can be different from the limit, or may not even exist.
- Example: $\displaystyle \lim_{x \to 2} (3x + 1) = 7$, because the outputs get close to 7 as $x$ gets close to 2.
- Example: $\displaystyle f(x) = \frac{x^2 - 4}{x - 2}$ is undefined at $x = 2$, but for $x \neq 2$ it equals $x + 2$, so $\displaystyle \lim_{x \to 2} f(x) = 4$.
Practice
What does $\displaystyle \lim_{x \to 3} f(x) = 5$ mean?
A limit describes the output values approached near $x = 3$. The value at 3 itself does not matter.
Find $\displaystyle \lim_{x \to 2} (3x + 1)$.
The outputs approach $3(2) + 1 = 7$.
Let $\displaystyle f(x) = \frac{x^2 - 4}{x - 2}$. Which statement is true?
For $x \neq 2$, $f(x) = x + 2$, which approaches 4. The limit ignores the value at 2.
Ready for more? Take the Unit 1 test.