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.
To find the limit as x approaches infinity and negative infinity for the function f(x) = cos(1/x), we need to consider the behavior of the function as x gets larger and larger (approaches infinity) or smaller and smaller (approaches negative infinity). Keep in mind that the value of cos(1/x) is only defined for nonzero values of x.
1. Limit as x approaches infinity:
To find the limit as x approaches infinity for f(x), we can analyze the function behavior by simplifying it. As x becomes large, 1/x approaches 0. Therefore, we can replace 1/x with 0 in the expression: f(x) = cos(1/x) ≈ cos(0) = 1. So, as x approaches infinity, the function f(x) approaches 1.
2. Limit as x approaches negative infinity:
Similarly, as x approaches negative infinity, we replace 1/x in the expression cos(1/x) with 0 since 1/x approaches 0 as x gets smaller. Thus, f(x) ≈ cos(0) = 1. Consequently, as x approaches negative infinity, the function f(x) also approaches 1.
So, both the limit as x approaches infinity and the limit as x approaches negative infinity for f(x) are equal to 1.
To identify horizontal asymptotes, we need to determine which values the function approaches as x becomes very large or very small. In this case, as we discussed earlier, the function approaches 1 as x approaches both positive infinity and negative infinity. Therefore, we have a horizontal asymptote at y = 1.