Question

Let f be a function defined for all real numbers. Which of the following statements must be true about f? Which might be true? Which must be false? Justify your answers.

(a) lim of f(x) as x approaches a = f(a)

(b) If the lim of f(x)/x as x approaches 0 = 2, then f(0)=0.

(c) If the lim of f(x)/x as x approaches 0 =1, then the lim of f(x) as x approaches 0 =0

(d) If the limit of f(x)-f(0) all over x as x approaches 0=3, then f'(0)=3.

Answer 1

(a) might be true (sin(0)/0 not defined)
(b) might be true (f(0) might not be defined)
(c) must be true
(d) must be true - that's the definition of f'