Limit Prove

[nycmathguy]

Calculus
Section 1.6

This prove question is unclear to me.

 

[MathLover1]

note: lim(f(x), x->0)=f(0)

let us find the following limits:
lim(f(x), x->0) and if x is a rational number, will be
lim(x^2,x->0 ) =0^2=0

lim(f(x)) as x->0 and if x is a irrational number, will be

Since the limits above are same, it proves that lim(f(x), x->0) does exist

 

[nycmathguy]

note: lim(f(x), x->0)=f(0)

let us find the following limits:
lim(f(x), x->0) and if x is a rational number, will be
lim(x^2,x->0 ) =0^2=0
View attachment 2586

lim(f(x)) as x->0 and if x is a irrational number, will be
View attachment 2583

Since the limits above are same, it proves that lim(f(x), x->0) does exist

Not too bad, honestly.