Question

Lim x`rarr` `oo` `(2-t+sint)/(t+cost` heelp me please?

Answer 1

The limit equals `-1`.

Explanation:

I'm first of all assuming that we want to find the limit as `t -> oo`. If so, we divide the entire expression by `t`.

`L = lim_(t->oo) (2/t - t/t + sint/t)/(t/t + cost/t)`

`L = lim_(t->oo) (2/t - 1 + sint/t)/(1 + cost/t`

The limit `lim_(t ->oo ) a/t` where `a` is an integer will always equal `0`. The limit `sint/t` and `cost/t` both equal `0` and have been derived using the squeeze theorem. For instance, because `-1 ≤ sinx ≤ 1` Dividing both sides by `x`, we see that `-1/x ≤ sinx/x ≤ 1/x`. We know that `lim_(x->oo) (-1/x) = 0` and `lim_(x->oo) 1/x = 0`, therefore, `lim_(x->oo) sinx/x` must be `0`. The same can be shown for the cosine function.

`L = (0 - 1 + 0)/(1 + 0)`

`L = -1`

We can check graphically.

Hopefully this helps!