CH 2: Limits

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

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24 Terms

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If Vertical Angle = 0 then…
Answer is: Infinity or negative infinity, 1/0 = infinity
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ln (0)
negative infinity
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e^0
1
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e^(negative infinity)
0
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e^(1/infinity)
0
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e^(infinity)
infinity
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1/inifnity
0
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“BDTN”

1. This is with growth rates
2. This is negative or positive infinity
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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
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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 \[-2,-1\]”
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lim tan(3x)/x x→0
3
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f(x)=2x^(2) - x, find lim f(x+h) - f(x)/h h→0
4x-1
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Average Rate of Change Formula
f(b) -f(a)/ b-a

given interval \[a,b\]
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Lim f(x) vs. f(0)

x→0 Actual Value

\
DNE of the actual value is more than the limit
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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
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First step to any limit:
PLUG IT IN (0/0 is a sign to factor)
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Special limits only when:
x→ 0;

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

lim 1-cosx/x = 0
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Two-sided limits
* Left (negative)
* Right (negative)
* Answer should be in y-value
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Greatest Integer Function:
Rounding Down.

Ex: y= II0.3II = 0
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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
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Discontinuity Types:
Removeable: Holes (VA and HA)

Non-removeable: Jump, Infinite, and Oscillation
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1/0
infinity
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1/0 from the left
negative infinity
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SOAP: Factor cubic polynomial
(a-b)(a^2+ab-b^2)

a= x

b= cubic root of second term in numerator