RES.18-005 | Spring 2010 | Undergraduate

Highlights of Calculus

Derivatives (12 videos)

Limits and Continuous Functions

What does it mean to say that a sequence of numbers a1, a2, …  approaches a LIMIT A ?
This means:  For any little interval around A, the numbers eventually get in there and stay there.

The numbers a1 = 1/2, a2 = 2/3, a3 = 3/4, …  approach the limit 1.   The first a’s DON’T MATTER
Change 2000 a’s and the limit is still 1.   What about powers of the a’s like a1^b1   a2^b2 …..  ??
If the b’s approach B then those powers approach A^B  except DANGER if B = 0 or infinity

For calculus the important case where you CAN’T TELL by just knowing A and B is A/B = 0/0
If f(x) and g(x) both get small  ( f/g looks like 0/0 ) then l’Hopital looks at slopes:  f/g goes like f ‘/g’

When is f(x) continuous at x=a ??  This means: f(x) is close to f(a) when x is close to a. See end of video

Professor Strang’s Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.

Course Info

As Taught In
Spring 2010