Prove Limit is 1/2

[nycmathguy]

Calculus
Section 2.4

Can you work out 36 in full?

 

[MathLover1]

36.

lim(1/x,x->2)=1/2

Consider |f(x)-L|<ϵ

|1/x-1/2|<ϵ

|(2-x)/(2x)|<ϵ
|2-x|<2xϵ
|-(x-2|< -2xϵ
-( -2xϵ)<|x-2|< -2xϵ
2xϵ<|x-2|< -2xϵ

since ϵ, choose -2xϵ
|x-2|<δ

then
|x-2|<-2xϵ
|x/(-2x)-2/(-2x)|<ϵ
|-1/2+1/x|<ϵ
|1/x-1/2|<ϵ ->proven

 

[nycmathguy]

36.

lim(1/x,x->2)=1/2

Consider |f(x)-L|<ϵ

|1/x-1/2|<ϵ

|(2-x)/(2x)|<ϵ
|2-x|<2xϵ
|-(x-2|< -2xϵ
-( -2xϵ)<|x-2|< -2xϵ
2xϵ<|x-2|< -2xϵ

since ϵ, choose -2xϵ
|x-2|<δ

then
|x-2|<-2xϵ
|x/(-2x)-2/(-2x)|<ϵ
|-1/2+1/x|<ϵ
|1/x-1/2|<ϵ ->proven

Wow! Impressive.