Proofs

Give the official definition of a Limit

lim f(x) = L

x→a

iff

For every E>0 (no matter how small) there exists d>0 such that when l x-a l <d then I f(x)-L l <E

Give three requirements to be continuous at x=a

1) f(a) exists

2) lim f(x) as x→a exists

3) f(a)= lim f(x) x→a

Using the definition of a limit prove lim x=a as x→a

let E>0, l x-a l <d and d=E

then l f(x)-L l

= l x-a l

<d=E

lim x=a as x→a

Prove lim [ f(x) + g(x) ] = lim f(x) + lim g(x)

Let lim f(x)=L as x→a and lim g(x)=M as x→a, E>0

For every E>O there exists d>0 such that when l x-a l <d then l f(x)-L l < E/2

let d= min {d, d2}

then l [ f(x) + g(x) ] - [ L+M ]

= l [ f(x)-L + g(x)-M ] l

= E/2 + E/2 = E

lim [ f(x)+g(x) ] = L+M = lim f(x) + lim g(x)