f(x) = cos (1/x)

Find the limit as x approaches infinity and find the limit as x approaches negative infinity.

Identify all horizontal asymptotes.

1/x --> 0 as x --> infinity

so, what is cos(0)?

To find the limit as x approaches infinity and negative infinity for the function f(x) = cos(1/x), we can use the properties of trigonometric functions and limits.

Limit as x approaches infinity:
As x approaches infinity, 1/x approaches 0. We know that the cosine function oscillates between -1 and 1 for all inputs. Since the input of our cosine function, 1/x, approaches 0 as x approaches infinity, the cosine of 1/x remains oscillating between -1 and 1 infinitely. Therefore, the limit as x approaches infinity for f(x) does not exist.

Limit as x approaches negative infinity:
Similarly, as x approaches negative infinity, 1/x approaches 0, and the cosine function still oscillates between -1 and 1 for all inputs. Therefore, the limit as x approaches negative infinity for f(x) does not exist as well.

Horizontal asymptotes:
To identify horizontal asymptotes, we check the behavior of the function as x approaches positive infinity and negative infinity. Since the limits do not exist in both cases, there are no horizontal asymptotes for the function f(x) = cos(1/x).