provide an epilson delta proof

Question

How would i provide a ε—δ proof for this

lim _x→2 5 = 5

Answer 1

So f(x) = 5.  The fundamental goal of an ε - δ proof is that given ε > 0, we must find a δ > 0 such that for any x such that |x - 2| < δ, we can show that |f(x) - 5| < ε.  In this case, for any ε > 0, we may arbitrarily choose δ = 1.  Then, we observe that for any x such that |x - 2| < δ = 1, we have |f(x) - 5| = |5 - 5| = 0 < ε.  Therefore we have lim_x→2f(x)=5.

 

Note that in this proof, the choice of δ can be arbitrary because we don't need to use the condition |x - 2| < δ to arrive at the conclusion, since our desired conclusion is true for all real values for x.