$\lim\limits_{t \to \infty} f'(t) = 0 \impliedby \lim\limits_{t \to \infty} f(t)=c $ is not true! Suppose $f(t)$ is continuously differentiable and $c$ is a finite constant. We know that
\begin{equation}
\lim\limits_{t \to \infty} f(t)=c \implies\lim\limits_{t \to \infty} f'(t) = 0   \quad \text{is NOT true!}.
\end{equation}
According to Barbalat's Lemma, this is true if $f'(t)$ is uniformly continuous. But can anyone give me a counterexample showing that $\lim\limits_{t \to \infty} f(t)=c$ NOT implying $\lim\limits_{t \to \infty} f'(t) = 0 $?

I take that $f'$ should exist. An interesting example is if: 
$$f(x) = \int_0^x \sin(t^2) \mathrm dt$$ 
Then: $$\lim_{x \to \infty} f(x) = \frac 1 2 \sqrt{\frac \pi 2}$$ 
but: $$\lim_{x \to \infty} f'(x) = \lim_{x \to \infty} \sin(x^2)$$ 
which doesn't exist. 
