Precisel Definition of Limit at Infinity

[nycmathguy]

Calculus
Section 2.6

See Definition 7 below.

 

[MathLover1]

Given ϵ =0.1 and ϵ=0.05

if x>N then |f(x)-L|<ϵ


|(1-3x)/sqrt(x^2+1)-(-3)|<0.1

|(1-3x)/sqrt(x^2+1)+3|<0.1

x>11.2826 => integer solutions x>12

|(1-3x)/sqrt(x^2+1)-(-3)|<0.05

|(1-3x)/sqrt(x^2+1)+3|<0.05

x>21.3791 => integer solutions x>22


since x>N, fist integer solution is N=11

 

[nycmathguy]

View attachment 3337


Given ϵ =0.1 and ϵ=0.05

if x>N then |f(x)-L|<ϵ


|(1-3x)/sqrt(x^2+1)-(-3)|<0.1

|(1-3x)/sqrt(x^2+1)+3|<0.1

x>11.2826 => integer solutions x>12

|(1-3x)/sqrt(x^2+1)-(-3)|<0.05

|(1-3x)/sqrt(x^2+1)+3|<0.05

x>21.3791 => integer solutions x>22


since x>N, fist integer solution is N=11

Thank you. Trust me, I will seriously study all your replies to my threads involving proofs when time allows.