Answer
$y=x+1$
Work Step by Step
Step 1. Given the function $y=x+x\cdot sin\frac{1}{x}$, we need to find all the oblique asymptotes.
Step 2. We need to find the end behavior or the limits of the function when $x\to\pm\infty$
Step 3. $\lim_{x\to\infty}y=\lim_{x\to\infty}(x+x\cdot sin\frac{1}{x})=\lim_{x\to\infty}(x+\frac{sin\frac{1}{x}}{1/x})$. As $x\to\infty, \frac{1}{x}\to0$, we have $\lim_{x\to\infty}(x+\frac{sin\frac{1}{x}}{1/x})=x+1$
Step 4. $\lim_{x\to-\infty}y=\lim_{x\to-\infty}(x+x\cdot sin\frac{1}{x})=\lim_{x\to-\infty}(x+\frac{sin\frac{1}{x}}{1/x})=x+1$
Step 5. Thus, the oblique asymptote of the function is $y=x+1$.
