proof of limit properties

They are basic properties without proof
concept of limit come from graphics
the proof is afterwards using words to describe
[ if for all ε > 0 , there exists S > 0 such that | f(x) − L | < ε whenever x > S ]

but we do not use proofs to learn limit

For (1), k is constant and independent of x, so the result is k

(2) can be deducted from (4), with g(x)=2 [ constant function ]

For (3),
f and g are both converging to limits L and M
thus, let h(x)=f(x)+g(x),
limit of h = limit of f + limit of g
limit of [ f(x)+g(x) ]= L+M
f(x)-g(x) is similar

(you may draw a graph to understand)


For (4),
when x approach to a, f and g become constants L and M
thus, f*g=L*M when x approach to a
i.e. lim{x tends to a} (f(x)*g(x))=L*M


(5) is similar to (4)

For (6),
when x approach to a, value of g(x) is M
thus, f( "g(x)" ) = f( M )
this result is only for f(x) is continuous at M
otherwise, it is indetermine

2012-01-24 22:47:32 補充：
you do not need the proof of properties
remember and understand properties is ok