Limit involving a Differential Equation 
Problem:
Consider the differential equation $$\dfrac{dy}{dx} + 10y = f(x) , \
> x>0,$$ where $f(x)$ is a continuous function such that $\displaystyle \lim_{x \to \infty} f(x) = 1$.
Find the value of $\displaystyle \lim_{x \to \infty} y(x)$.

My attempt:
Since this is a Linear Differential Equation, we can solve for $y(x)$ to get,
$$ y(x) = \dfrac{C + \int e^{10x} f(x)}{e^{10x}} $$
which gives,
$$ \require{cancel} \lim_{x \to \infty} y(x) = \cancelto{0}{\lim_{x \to \infty} \dfrac{C}{e^{10x}}} + \lim_{x \to \infty}\dfrac{{\int e^{10x} f(x)}}{e^{10x}} =  \lim_{x \to \infty}\dfrac{{\int e^{10x} f(x)}}{e^{10x}} $$
But I don't know how to solve for the last limit involving the integral.

$$y(x)=\frac{ \int _1^xe^{10 t} f(t)\,dt}{e^{10x}}+C e^{-10 x}$$
$$\lim_{x\to\infty}\frac{ \int _1^xe^{10 t} f(t)\,dt}{e^{10x}}\to\frac{\infty}{\infty}$$
use L'Hopital rule.
$$\frac{d}{dx}\int _1^xe^{10 t} f(t)\,dt=e^{10x}f(x)$$
so we have
$$\lim_{x\to\infty}\frac{e^{10 x} f(x)}{10e^{10x}}=\frac{1}{10}$$
