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Determine whether given statements about a function differentiable at a point are true or false

Let f be a function and a a point such that f'(a) exists. Determine which of the following are true or false.

  1. \displaystyle{f'(a) = \lim_{h \to a} \frac{f(h) - f(a)}{h-a}}.
  2. \displaystyle{f'(a) = \lim_{h \to 0} \frac{f(a) - f(a -h)}{h}}.
  3. \displaystyle{ f'(a) = \lim_{t \to 0} \frac{f(a+2t) - f(a)}{t}}.
  4. \displaystyle{ f'(a) = \lim_{t \to 0} \frac{f(a+2t) - f(a+t)}{2t}}.

  1. True. Let k = h-a. Then k \to 0 as (h-a) \to 0 implies k \to 0 as h \to a. Thus,

        \[ f'(a) = \lim_{k \to 0} \frac{f(a+k) - f(a)}{k} = \lim_{h \to a} \frac{f(a+h-a) - f(a)}{h-a} = \lim_{h \to a} \frac{f(h) - f(a)}{h-a}. \qquad \blacksquare \]

  2. True. Let k = -h, then

        \[ \lim_{h \to 0} \frac{f(a) - f(a-h)}{h} = \lim_{k \to 0} \frac{f(a) - f(a+k)}{-k} = \lim_{k \to 0} \frac{f(a+k) - f(a)}{k} = f'(a). \qquad \blacksquare \]

  3. False. First, from the definition of limit we can conclude

        \[ \lim_{t \to 0} f(t) = \lim_{2t \to 0} f(t) \]

    (by taking \delta' = \frac{\delta}{2}). Then we have,

        \begin{align*}  \lim_{t \to 0} \frac{f(a+2t) - f(a)}{t} &= \lim_{2t \to 0} \frac{f(a+2t) - f(a)}{t} \\  &= 2 \lim_{2t \to 0} \frac{f(a+2t) - f(a)}{2t} \\  &= 2 f'(a) \neq f'(a) & \text{if } f'(a) \neq 0. \end{align*}

    Thus, this is not true in general (since there are many function differentiable at a point a such that the derivative at that point is nonzero). ( Note: This statement would be true were the denominator in the given equation 2t instead of t.)

  4. False. Here we evaluate,

        \begin{align*}  \lim_{t \to 0} \frac{f(a+2t) - f(a+t)}{2t} &= \frac{1}{2} \left( \lim_{t \to 0} \frac{f(a+2t)-f(a)}{t} - \lim_{t \to 0} \frac{f(a+t) - f(a)}{t} \right) \\  &= \frac{1}{2} ( 2f'(a) - f'(a)) \qquad (\text{part (c)})\\  &= \frac{1}{2} f'(a) \neq f'(a) \qquad \text{if } f'(a) \neq 0. \end{align*}

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