# 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.

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