Can you find a number delta so that when x -0 lt delta fx

Question

For problems 1-5, prove the statement using the ∈-δ definition of a limit:

(1) lim_x→4 (2+1/4x) = 3

(2) lim_x→-1 (3x + 5) = 2

(3) lim_x→-1 (1 +4x)/3 = -1

(4) lim_x→0 x⁴ = 0

(5) lim_x→4 x² = 16(Hint: If|x-4|<1, What can you say about |x+4|?)

Conceptual: Consider the function

f(x) = |x|/25x

We want to try to prove the following statement:

lim_x→0 f(x) = 1/25

(6) We start with ∈ = 1/2. Can you find a number δ so that when | x -0| < δ, |f(x) - 1/25|< 1/2

(7) We now have some reason to believe that the above statement in true. But having "some reason to believe" isn't enough for mathematicians. Repeat the previous problem for ∈ = 1/10.

(8) Try the limit one more time, this time for ∈ = 1/100 . Make sure that when you are trying to verify the condition |x - 0| < δ, you check both positive and negative values for x. What do you think about the limit we are trying to prove?