Calculus: Limit & Continuity

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limits

20 Terms

1

as f(x) x approaches c, it is equal to L

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2

When does limit exist?

Both right and left handside limits are equal to each other.

<p>Both right and left handside limits are equal to each other. </p>
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3

Finding Limits End Behavior

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4

Which value makes vertical asymptotes exist

The value makes denominator (분모) equals to zero.

<p>The value makes denominator (분모) equals to zero.</p>
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5
<p>Finding Horizontal Asymptote</p>

Finding Horizontal Asymptote

y = 2/3

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6
<p>Finding Horizontal Asymptote</p>

Finding Horizontal Asymptote

y = 0

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7
<p>Finding Horizontal Asymptote</p>

Finding Horizontal Asymptote

No Horizontal Asymptotes

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8

Squeeze Theorem

Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) for all x in an interval except possibly at x = c, and lim f(x) = lim h(x) = L, then lim g(x) = L.

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9

How to find limits involving infinity?

We need to divide by the biggest power.

<p>We need to divide by the biggest power. </p>
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10

Simplified Version of finding limits involving infinity

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11

As x approaches 0, limit of sin x/x is

lim (x -> 0) sin(x)/x = 1

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12

As x approaches infinity, limit of sin x/x is

lim (x -> ∞) sin(x)/x = 0

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13

Definition of Continuity

No holes, breaks, or jumps

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14

Three conditions to satisfy that y = f(x) is continuous at x = c.

1) f(c) exists

2) lim (x → c) f(x) exists

3) lim (x → c) f(x) = f(c)

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15

Jump Discontinuity

lim (x→c+) f(x) ≠ lim (x→c-) f(x)

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16

Removable Discontinuity

lim (x→c) f(c) ≠ f(c)

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17

Infinite Discontinuity

When there’s Asymptotes

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18

Theorems on Continuous Functions

1. The Extreme Value Theorem

Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f has both a maximum and minimum value on [a, b].

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19

Theorems on Continuous Functions

1. The Intermediate Value Theorem

If ( f ) is continuous on ([a,b]) and ( k ) is between ( f(a) ) and ( f(b) ), then there exists a ( c ) in ((a,b)) such that ( f(c) = k ).

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20

Example of The Intermediate Value Theorem

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