Finding N in epsilon-delta proof for lim(x→-∞) (-5x/x-2)=-5

[tesla93]

The problem is:

lim as x→- ∞ (-5x/x-2)=-5

And I have to find the best interval of solutions for N

Relevant equations:
if x>N then |(-5x/x-2)-(-5)|<ε

My attempt:

|(-5x/x-2) - (-5)|<ε

|-10/x-2|<ε

x-2< 10/ε

as x→-∞ we can assume that x-2<0

x-2<10/ε

x< (10/ε)+2

therefore, the interval should be NE [(10/ε) +2, -∞)

however, the answer says that it should be -2 not +2. I don't know what I am doing wrong, if anyone can help that would be great! Thanks!

 

[LCKurtz]

The problem is:

lim as x→- ∞ (-5x/x-2)=-5

Of course, you should write that fraction as -5x /(x-2) here. Use parentheses!

And I have to find the best interval of solutions for N

Relevant equations:
if x>N then |(-5x/x-2)-(-5)|<ε

My attempt:

|(-5x/x-2) - (-5)|<ε

|-10/x-2|<ε

At this step you have 10/|x-2| < ε

x-2< 10/ε

No. You should have |x-2| > 10/ε. You need absolute value signs and the inequality is reversed.

as x→-∞ we can assume that x-2<0

x-2<10/ε

If x - 2 < 0, what is |x-2|?