Problem: Prove or provide a counterexample: Let $m$ be the Lebesgue measure on $\mathbb{R}$. Suppose $f\in L^1(\mathbb{R}, m)$ is of class $C^1$, and that $f'\in L^1 (\mathbb{R}, m)$. Then $\lim_{x\rightarrow \infty} f(x) =0$.
Thoughts: Take the standard Gaussian function that integrates to 1. By changing the "variance" (using $e^{-x^2/c}$ for some $c$) we can vary the total integral while keeping the maximal value as 1. Translating these across the real line so that we have functions centered on each positive integer $n$ with total integral $\frac{1}{n^2}$ (and adding them together and using the monotone convergence theorem), we get a $C^\infty$ function in $L^1$. My scratchwork indicates the derivative is also in $L^1$, so this seems to provide a counterexample.
However, upon further review, my scratch work is incorrect...