Question

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

Answer 1

`lim_(xrarroo)(1/2x+sinx) = oo`

Explanation:

Explanation 1

As `x` increases without bound, so does `1/2x`. Meanwhile `sinx` 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_(xrarroo)sinx/x = 0` (Use the squeeze theroem.)

`1/2x+sinx = x(1/2+sinx/x)`

`lim_(xrarroo)(1/2x+sinx) = lim_(xrarro)(x(1/2+sinx/x))`

The first factor goes to `oo` and the second to `1/2`, so the product goes to `oo`.