Limits and Continuity - AP Calculus Unit 1 Review

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Timestamps
0:00 Intro
0:45 Intro to Continuity
6:19 Naming Discontinuities
8:47 Definition of Continuity
11:44 Intuition for Limits
16:38 Limits from the Left & Right
19:20 Limits and Continuity

[sardineman]

I like the way you explained very easy to understand and the second black pin you used was very nice

[sardineman]

Lim x-> infinity boy sin(x)/x

-1 <|= sin(x) <|= 1 the range of sin(x) is [-1, 1]

-1/x <|= sin(x)/x <|= 1/x dividing them does not change the inequality signs because positive boy always more than negative boy

Lim x-> infinity boy -1/x <|= Lim x-> infinity boy sin(x)/x <|= Lim x-> infinity boy 1/x we took the limit boy of all of them

0 <|= Lim x-> infinity boy sin(x)/x <|= 0 I think we can agree that any number divided by infinity is zero for limits that is
Our limit is between the same number which means it is that number because of how equality works with real numbers

Lim x-> infinity boy sin(x)/x =0

[sardineman]

Lim x-> 0 sin(x)/x
=lim x->0 x/x
=1
Life is so much easier as an engineer :)

[sardineman]

Lim x->0 sin(x)/x know: <|= means less than or equal to & >|= you can guess

Sin(x) <|= x <|= tan(x), x€ [0, pi/2) everything is positive over this interval

1/Sin(x) >|= 1/x >|= cos(x)/sin(x) inverting them all flipped the inequality

1 >|= sin(x)/x >|= cos(x) I multiplied them all by sin(x) this does not change the inequality sign

Lim x->0 1 >|= Lim x->0 sin(x)/x >|= Lim x->0 cos(x) we have taken the same limit for all of them

1 >|= lim x->0 sin(x)/x >|= 1 now we see you that the limit is in between the same number which means it must be that number because real number stuff
lim x->0 sin(x)/x =1 ●