AP Calculus BC / Unit 1: Limits and Continuity
1.2 Defining Limits and Using Limit Notation
Notes
$\displaystyle \lim_{x \to c} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to $c$ (from both sides), whether or not $f(c)$ exists.
- One-sided limits: $\displaystyle \lim_{x\to c^-}$ (from the left) and $\displaystyle \lim_{x\to c^+}$ (from the right).
- The two-sided limit exists only when both one-sided limits exist and are equal.
Practice
If $\displaystyle \lim_{x\to 2^-} f(x)=5$ and $\displaystyle \lim_{x\to 2^+} f(x)=5$, then:
Equal one-sided limits mean the two-sided limit exists. It says nothing about $f(2)$.
If $\displaystyle \lim_{x\to 1^-} g(x)=3$ and $\displaystyle \lim_{x\to 1^+} g(x)=-1$, then $\displaystyle \lim_{x\to 1} g(x)$:
The one-sided limits disagree.
The notation $\displaystyle \lim_{x\to 4^+} h(x)$ means the limit as $x$ approaches 4:
The plus sign means approaching from values above 4.
Ready for more? Take the Unit 1 test.