25.
A function f is said to have a limit L as x approaches c, denoted
View attachment 2706 , if for every ε>0, there exists a δ>0 such that |x−c|<δ implies |f(x)−L|<ε.
Then, to prove that
View attachment 2707 , we must show that for any ε>0 there exists δ>0 such that |x−0|<δ implies
View attachment 2708.
Proof:
Let ε>0 be arbitrary, and let δ=min{ε,1}.
Suppose |x−0|=|x|<δ. Note that as δ≤1, we have |x|<1, meaning
View attachment 2709. Furthermore, as δ≤ε, we have |x|<ε. With those facts:
View attachment 2710
Thus, for an arbitrary ε>0, we have found a δ>0 such that |x−0|<δ implies
, meaning
View attachment 2711
.