Limit at infinity of a combined trig/rational function

As $x \to +\infty$, $\frac{1}{x} \to 0$, so $\cos\left(\frac{1}{x}\right) \to \cos(0) = 1$. This handles the first term.

For the second term, divide numerator and denominator by $x$: $\frac{3x}{x-1} = \frac{3}{1 - 1/x} \to \frac{3}{1-0} = 3$ as $x \to +\infty$. Adding the two results gives $1 + 3 = 4$, so the limit is $4$ and $y = 4$ is a horizontal asymptote of $g(x)$ as $x \to +\infty$.

The limit computed in part a was four, since the cosine term approached one and the rational term approached three. Adding these gives four. Therefore g of x approaches the number four as x approaches positive infinity, matching the horizontal asymptote y equals four.