Calculus: DirectionalLimits by: javier

Calculus: Directional Limits 15 10 5 0 −5 0

2

4

by: javier

6

8

10

Directional Limits an intuitive approach

Directional Limits: an intuitive approach

IF □ (right limit exists) lim f(x) = L

x→3+

□ (left limit exists) lim f(x) = M

x→3−

□ (they are equal) L=M

THEN we say the limits eixsts, we write lim f(x) = L

x→3

Directional Limits: an intuitive approach

□

6

□ 4

lim f(x)

x→0+

lim f(x)

x→0−

□ lim f(x) x→0

2

0 −4

−2

0

2

4

Directional Limits: an intuitive approach

15

□

10

□

5

□

0

□

−5 0

2

4

6

8

10

□

lim f(x)

x→0+

lim f(x)

x→4+

lim f(x)

x→4−

lim f(x)

x→8−

lim f(x)

x→8+

Directional Limits: an intuitive approach

3

□

2

□ □

1

□

0

□

f(x) = ⌊x⌋

−1 −1

lim f(x)

x→0+

lim f(x)

x→2+

lim f(x)

x→2−

lim f(x)

x→3−

lim f(x)

x→3+

□ lim f(x) 0

1

2

3

4

x→3

□ lim f(x) x→2.5

Directional Limits: an intuitive approach

1

□ lim f(x)

0

□

x→∞

□

−1

□

−2 −3

f(x) = ln(x) 0

0.5

1

1.5

2

lim f(x)

x→0+

lim f(x)

x→1+

lim f(x)

x→1−

Directional Limits: an intuitive approach

1

□

0.5

□ □

0

□

−0.5 f(x) = sin

−1 −0.5

0

0.5

(1) x

1

lim f(x)

x→0−

lim f(x)

x→0+

lim f(x)

x→ π2 +

lim f(x)

x→π −

Directional Limits: an intuitive approach

Algebraic Example: study the following limit

|x| x→0 x lim