Question

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\cos \left(\frac{1}{x}\right)$$

Short answer

The limits as \(x\) approaches infinity and negative infinity do not exist as the function oscillates between -1 and 1. The horizontal asymptotes for the function are \(y = 1\) and \(y = -1\).

Step-by-step solution

Understanding the behaviour of the function

The function \(f(x)=\cos \left(\frac{1}{x}\right)\) becomes more oscillatory as \(x\) approaches 0. Cosine function oscillates between -1 and 1, hence for large \(x\), the value of \(f(x)\) will oscillate between -1 and 1.

Finding the limit as \(x\) approaches infinity

As \(x\) becomes larger, \(\frac{1}{x}\) becomes closer to 0. Hence, the function tends to oscillate faster and faster. Thus, the limit as \(x\) approaches infinity is undefined because the function does not approach a particular value but oscillates between -1 and 1.

Finding the limit as \(x\) approaches negative infinity

Similarly, as \(x\) becomes infinitely negative, our function also oscillates between -1 and 1 without settling down to a particular number. So the limit as \(x\) approaches negative infinity is also undefined.

Identify horizontal asymptotes

A horizontal asymptote is a horizontal line that the curve approaches as \(x\) tends to infinity or negative infinity. Since function \(f(x)\) oscillates between -1 and 1, the lines \(y = 1\) and \(y = -1\) are the horizontal asymptotes.