EDB — 1CX

Exercises

1. [1CX]Let f be differentiable in the interval \((a, b)\), let \(x_ 0 ∈ (a, b)\) and \(x_ 0 {\lt} 𝛼_ n {\lt} 𝛽_ n , 𝛽_ n → x_ 0\) for \(n →∞\). Show that if the sequence \(\frac{𝛽_ n - x_ 0}{𝛽_ n - 𝛼_ n}\) is bounded then

   \[ \frac{f (𝛽_ n ) - f (𝛼_ n )}{𝛽_ n - 𝛼_ n}→_ n f' (x_ 0 ) \]

   Show by example that this conclusion is false if the given condition is not verified. [ [1CY]↺↻]