Finding delta and epsilon for a funtion

Question

Let f(x)=3x+5. Find the largest delta so that the absolute value of f(x)-f(a) is less than epsilon when the absolute value of x-a is less than delta

Answer 1

Let δ = ε/3.  

 

Then if l x - a l < δ = ε/3, we have -ε/3 < x - a < ε/3

 

So. -ε < 3x - 3a < ε

 

      -ε < (3x+5) - (3a+5) < ε

 

      -ε < f(x) - f(a) < ε

 

That is, l f(x) - f(a) l < ε