CH 2: Limits

Includes: 2.2, 2.3, 2.4, 2.5, 4.5, and 8.7

If Vertical Angle = 0 then…

If Vertical Angle = 0 then…

Answer is: Infinity or negative infinity, 1/0 = infinity

ln (0)

negative infinity

e^0

1

e^(negative infinity)

0

e^(1/infinity)

0

e^(infinity)

infinity

1/inifnity

0

“BDTN”

1. This is with growth rates

2. This is negative or positive infinity

Continuity:

1. Left and right limit are equal

2. lim f(x) = f(c) for all x=c therefore (triangle dots) f(x) is cont. x→c

IVT (intermediate value form): Answer formats

1. lim f(x) = f(c) for all x=c therefore (triangle dots) f(x) is cont. x→c (only if polynomial; 1/x is not cont.)

2. Plug interval into polynomial (Answers must be have a range large enough to include number you are trying to hit)

3. “Therefore (triangle dots) by IVT f(x) has a solution to f(x)=1 on the [-2,-1]”

lim tan(3x)/x x→0

3

f(x)=2x^(2) - x, find lim f(x+h) - f(x)/h h→0

4x-1

Average Rate of Change Formula

f(b) -f(a)/ b-a

given interval [a,b]

Lim f(x) vs. f(0)

x→0 Actual Value

DNE of the actual value is more than the limit

Reasons Limits DNE:

• lim IxI/x = approaches different numbers

  x→0

• lim 1/x^(2) = infinity

  x→0

• lim sin 1/x = oscillation

  x→0

First step to any limit:

PLUG IT IN (0/0 is a sign to factor)

Special limits only when:

x→ 0;

Lim sin(x)/x = 1 lim x/sinx =1 lim sin4x/4 = 1

lim 1-cosx/x = 0

Two-sided limits

• Left (negative)

• Right (negative)

• Answer should be in y-value

Greatest Integer Function:

Rounding Down.

Ex: y= II0.3II = 0

Infinity:

For limits with infinity divide every term with the highest power of x then simplify (Shortcut: HA Rules/ End Behavior)

Big on top: DNE

Big on Bottom: y=0

Equal= divide

Discontinuity Types:

Removeable: Holes (VA and HA)

Non-removeable: Jump, Infinite, and Oscillation

1/0

infinity

1/0 from the left

negative infinity

SOAP: Factor cubic polynomial

(a-b)(a^2+ab-b^2)

a= x

b= cubic root of second term in numerator