# How do you find lim_(xtooo)(Cos(x)/x)?

Mar 28, 2017

Because the upper bound of cos(x) is 1 and the lower bound is -1:

${\lim}_{x \to \infty} - \frac{1}{x} \le {\lim}_{x \to \infty} \cos \frac{x}{x} \le {\lim}_{x \to \infty} \frac{1}{x}$

We know the limits of the two bounds to be 0:

$0 \le {\lim}_{x \to \infty} \cos \frac{x}{x} \le 0$

$\therefore$

${\lim}_{x \to \infty} \cos \frac{x}{x} = 0$

Mar 28, 2017

It should tend to zero.

#### Explanation:

I would say that as cos(x) oscilates between $1 \mathmr{and} - 1$ then $x$ will grow so much that the fraction will become very small tending to become zero.
This is somewhat supported by the graph of our function:

graph{(cos(x))/x [-10, 10, -5, 5]}

Because $- 1 \le \cos x \le 1$ , we have that

$- \frac{1}{x} \le \cos \frac{x}{x} \le \frac{1}{x}$

Hence

${\lim}_{x \to \infty} \left(- \frac{1}{x}\right) \le {\lim}_{x \to \infty} \cos \frac{x}{x} \le {\lim}_{x \to \infty} \frac{1}{x}$

$0 \le {\lim}_{x \to \infty} \cos \frac{x}{x} \le 0$

Finally ${\lim}_{x \to \infty} \cos \frac{x}{x} = 0$