Is there a way to get at this limit problem algebraically?

Question

$$\lim \limits_{x \to \infty} \frac{(1-x)}{\cos(x)} $$

Is there a way to get at this probelm algebrically? L^Hopital's Rule does not work here......sadly. The answer I got graphing it is that it does not exist.

Answer 1

The limit doesn't exist. If it did, we could take it along any sequence of points which go to infinity. This is typically how people define the limit of a function. It is the limit of a sequence of inputs, and the limit exists if every sequence of inputs converging to some point converges to the same limit.

But we can choose one sequence of points for which cosine is always positive, like say $x_n = 2n\pi$, and $f(x_n) \rightarrow-\infty$ along this sequence, and we can also choose a sequence of points for which cosine is always negative, like $y_n = (2n+1)\pi$, so that $f(y_n) \rightarrow \infty$ along this other sequence.

So we've produced two different sequences with different limits, so the limit doesn't exist.