Limits Graphically Slides
WARM-UP 01
Limits Graphically
Function \(y = f(x)\)
Graph of \(y = f(x)\) Focus on \(x = 3\)
x y -3 -2 -1 1 2 3 4 5 6 -1 1 2 3 y = f(x)
(a) \(\lim_{x \to 3^-} f(x) =\)
(b) \(\lim_{x \to 3^+} f(x) =\)
(c) \(\lim_{x \to 3} f(x) =\)
(d) \(f(3) =\)
(e) \(\lim_{x \to -\infty} f(x) =\)
(f) \(\lim_{x \to \infty} f(x) =\)
Examine one-sided limits vs. the isolated point value at \(x = 3\) Problem 1 of 6 →
WARM-UP 02
Limits Graphically
Function \(y = F(x)\)
Graph of \(y = F(x)\) Focus on \(x = 3\)
x y -3 -2 -1 1 2 3 4 5 1 2 3 y = F(x)
(a) \(\lim_{x \to 3^-} F(x) =\)
(b) \(\lim_{x \to 3^+} F(x) =\)
(c) \(\lim_{x \to 3} F(x) =\)
(d) \(F(3) =\)
(e) \(\lim_{x \to -\infty} F(x) =\)
(f) \(\lim_{x \to \infty} F(x) =\)
Compare the limit approaching the hole to the value of the solid point Problem 2 of 6 →
WARM-UP 03
Limits Graphically
Function \(y = F(x)\)
Graph of \(y = F(x)\)
x y -4 -3 -2 -1 1 2 3 1 2 3 y = F(x)
(a) \(\lim_{x \to -2^-} F(x) =\)
(b) \(\lim_{x \to -2^+} F(x) =\)
(c) \(\lim_{x \to -2} F(x) =\)
(d) \(F(-2) =\)
(e) \(\lim_{x \to -\infty} F(x) =\)
(f) \(\lim_{x \to \infty} F(x) =\)
Notice the jump gap and the separated point at \(x = -2\) Problem 3 of 6 →
WARM-UP 04
Limits Graphically
Function \(y = \phi(x)\)
Graph of \(y = \phi(x)\) Vertical Asymptote at \(x = -2\)
x y -4 -3 -2 -1 1 2 3 4 1 2 3 x = -2 y = φ(x)
(a) \(\lim_{x \to -2^-} \phi(x) =\)
(b) \(\lim_{x \to -2^+} \phi(x) =\)
(c) \(\lim_{x \to -2} \phi(x) =\)
(d) \(\phi(-2) =\)
(e) \(\lim_{x \to -\infty} \phi(x) =\)
(f) \(\lim_{x \to \infty} \phi(x) =\)
Track unbounded behavior approaching the vertical asymptote line Problem 4 of 6 →
WARM-UP 05
Limits Graphically
Function \(y = f(x)\)
Graph of \(y = f(x)\) Vertical Asymptote at \(x = 3\)
x y -3 -2 -1 1 2 3 4 5 1 2 -1 -2 x = 3 y = f(x)
(a) \(\lim_{x \to 3^-} f(x) =\)
(b) \(\lim_{x \to 3^+} f(x) =\)
(c) \(\lim_{x \to 3} f(x) =\)
(d) \(f(3) =\)
(e) \(\lim_{x \to -\infty} f(x) =\)
(f) \(\lim_{x \to \infty} f(x) =\)
Both branches plunge in the same direction: does the limit equal \(-\infty\)? Problem 5 of 6 →
WARM-UP 06
Limits Graphically
Existence of Limits
Graph of \(y = f(x)\) Test: \(x = -3, -1, 0, 2\)
x y -4 -3 -2 -1 0 1 2 3 1 2 3 4 y = f(x)
(a) \(\lim_{x \to -3} f(x) =\)
(b) \(\lim_{x \to -1} f(x) =\)
(c) \(\lim_{x \to 0} f(x) =\)
(d) \(\lim_{x \to 2} f(x) =\)
Does the limit exist when \(x = -3, -1, 0,\) and \(2\)?
Remember: A limit exists if and only if left-hand limit equals right-hand limit. Problem 6 of 6 • Warm-Up Complete
