Continuity from a Calculus Perspective
Learning Objectives
- The Calculus definition of continuity
- Determining continuity graphically
- Determining continuity algebraically
- Continuity of piecewise-defined functions
Exercises
1.
\(x=8\quad \) since \(f(8)\) is undefined.
\(x=-3\quad \) since \(\displaystyle \lim_{x\rightarrow -3} f(x)\) DNE.
\(x=3\quad \) since \(\displaystyle \lim_{x\rightarrow 3} f(x)\) DNE.
\(x=8\quad \) since \(\displaystyle \lim_{x\rightarrow 8} f(x)\neq f(8)\).
\(x=-3\quad \) since \(\displaystyle \lim_{x\rightarrow -3} f(x)\) DNE.
\(x=3\quad \) since \(\displaystyle \lim_{x\rightarrow 3} f(x)\) DNE.
\(x=8\quad \) since \(\displaystyle \lim_{x\rightarrow 8} f(x)\neq f(8)\).
2.
\((-\infty, -4)\cup (-4,4) \cup (4,\infty)\)
3.
\(\left[-7,\dfrac{19}{3}\right)\cup \left( \dfrac{19}{3}, \dfrac{20}{3}\right)\)
4.
\((-\infty, 2) \cup (2,8)\cup (8,9)\cup(9,\infty)\)
5.
\(b=-\dfrac{46}{5}\)