Linear Approximation Theorem: Proof of Differentability - Prof. E. R. Heal, Study notes of Mathematics

Math 4200

Linear Approximation Theorem:

Suppose is defined in an open interval containing Then is differentiable0 B Þ 0

at if and only if there exists a linear affine) functionB Ð

that approximates near in the sense thatEÐBÑ œ +ÐB  B Ñ  0 ÐB Ñ 0 B

lim

I. Suppose is differentiable at . Let .0 B œ B EÐBÑ œ 0 ÐB ÑÐB  B Ñ  0 ÐB Ñ

= 0 So,lim lim

= 0 .lim

II. Suppose there exists such thatEÐBÑ œ +ÐB  B Ñ  0ÐB Ñ

= 0 ,lim