Let \(\varphi\in C^1(\mathbb R)\) and suppose \[ \varphi(x)\to a, \qquad \varphi'(x)\to b \qquad(x\to\infty). \] Prove or give a counterexample: must \(b=0\)?
Solution
1. For every \(x\in\RR\), there is a point\[ \xi_x\in(x,x+1) \]such that\[ \varphi(x+1)-\varphi(x)=\varphi'(\xi_x). \]
Apply the mean value theorem to \(\varphi\) on the interval \([x,x+1]\).
2. One has\[ b=0. \]
As \(x\to\infty\), \[ \varphi(x+1)-\varphi(x)\longrightarrow a-a=0. \] Also \(\xi_x>x\), so \[ \xi_x\longrightarrow\infty. \] The hypothesis \(\varphi'(y)\to b\) as \(y\to\infty\) therefore gives \[ \varphi'(\xi_x)\longrightarrow b. \] Taking limits in the identity from step 1 yields \(0=b\).
3. Thus the answer is\[ \boxed{\text{yes: }b=0}. \]
This is exactly step 2.
4. Q.E.D.
Step 3 answers the question.