lim

Question

x---> 2 4x+1/3x-4

illustrate definition 2 by finding values of delta that correspond to epsilon=0.5 and epsilon= 0.1

I having trouble setting this up.
do i set up two different functions then divide?
0.2898 that's the answer i got....

Answer 1

clearly the limit is (8+1)/(6-4) = 9/2

Not sure what your definition 2 is, but if you need to find δ such that |f(x+δ)-9/2| < 0.5 then that means

|(4(2+δ)+1)/(3(2+δ)-4)-9/2| < 0.5
Work your way through that and you will find that δ < 0.125

So, if you pick a value of x within 1/8 of 2, f(x) will be within 1/2 of 9/2.

Or, more mathematically,

if |x-2| < 1/8, |f(x)-9/2| < 1/2