Analysis Problems #2 1. Let f be a function which is defined on the interval [0, 2]. Show that inf f (x) ≥ inf f (x). 0≤x≤1 0≤x≤2 2. As we are going to prove in class, limits of polynomial functions can be computed by simple substitution. For instance, one has lim (2x2 − 4x + 1) = 2 · 32 − 4 · 3 + 1 = 7. x→3 Assuming this fact for the moment, compute each of the following limits: 2x3 − 5x − 6 , x→2 x−2 lim x3 − 3x + 2 . x→1 (x − 1)2 lim 3. Let f be the function defined by { f (x) = 7 − 2x 3x − 3 if x ∈ Q if x ∈ /Q } . Use the ε-δ definition of limits to show that lim f (x) = 3. x→2 4. Evaluate the limit lim f (x) when f is the function defined by x→1 { f (x) = 1 + 3x 5 if x ̸= 1 if x = 1 5. Use the ε-δ definition of limits to show that lim x2 = 4. x→2 } .