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$f(x)$ is a differentiable function on the real line such that $\lim_{x \rightarrow \infty} f(x)=1$ and $\lim_{x \rightarrow \infty} f~'(x)=\alpha$. Then :

$(A) ~\alpha $ must be $0$

$(B) ~\alpha $ need not be $0$ but $|\alpha|<1$

$(C) ~\alpha > 1$

$(D) ~\alpha < -1$

Attempt:

$$f~'(x) = \lim_{h \rightarrow 0} \dfrac {f(x+h)-f(x)}{h}$$

$$\lim_{x \rightarrow \infty} f~'(x) =\lim_{x \rightarrow \infty} \lim_{h \rightarrow 0} \dfrac {f(x+h)-f(x)}{h}$$

$$\lim_{x \rightarrow \infty} f~'(x) = \lim_{h \rightarrow 0} \lim_{x \rightarrow \infty}\dfrac {f(x+h)-f(x)}{h}\tag{1}$$

$$\lim_{x \rightarrow \infty} f~'(x) = \lim_{h \rightarrow 0}\dfrac {1-1}{h} = 0 $$

Hence $\alpha$ must be equal to $0$.

Is step $(1)$ legal?

I mean, can we interchange the two limits?

When can we not interchange the two limits and when can we?

Thank you very much for your help in this regard.

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MathMan
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    The conclusion is correct: $;\alpha =0;$, yet step (1) needs justification and, in the general case, it can fail big time as you can see in some examples here: http://math.stackexchange.com/questions/15240/when-can-you-switch-the-order-of-limits – Timbuc Apr 29 '15 at 06:59

1 Answers1

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Justify each of the following steps:

We can try the following: for $\;n\in\Bbb N\;$ :

$$f(n+1)-f(n)=\frac{f(n+1)-f(n)}{(n+1)-n}=f'(c_n)\;\;,\;\;\;c_n\in (n,\,n+1)$$

using the MVT (Lagrange's Theorem) for differentiable functions.

From this it follows that $\;c_n\xrightarrow[n\to\infty]{}\infty\;$, and thus

$$\alpha=\lim_{n\to\infty}f'(c_n)=\lim_{n\to\infty}\left(f(n+1)-f(n)\right)=1-1=0$$

Timbuc
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