# What is the limit of (1/2x+sinx) as x goes to infinity?

Oct 22, 2015

${\lim}_{x \rightarrow \infty} \left(\frac{1}{2} x + \sin x\right) = \infty$

#### Explanation:

Explanation 1

As $x$ increases without bound, so does $\frac{1}{2} x$. Meanwhile $\sin x$ is bounded by $- 1$ and $1$. We have an increasing quantity plus a bounded quantity. The sun increases without bound.

Explanation 2

Note that ${\lim}_{x \rightarrow \infty} \sin \frac{x}{x} = 0$ (Use the squeeze theroem.)

$\frac{1}{2} x + \sin x = x \left(\frac{1}{2} + \sin \frac{x}{x}\right)$

${\lim}_{x \rightarrow \infty} \left(\frac{1}{2} x + \sin x\right) = {\lim}_{x \rightarrow o} \left(x \left(\frac{1}{2} + \sin \frac{x}{x}\right)\right)$

The first factor goes to $\infty$ and the second to $\frac{1}{2}$, so the product goes to $\infty$.