Understanding limits is crucial in calculus, as they represent the value a function approaches as the input approaches a specific number. When evaluating limits, we often consider the behavior of a function as it approaches a point from both the left and right sides on the x-axis. However, limits can also involve infinity, either as an output or an input. This summary focuses on the latter, exploring how functions behave as the input approaches positive or negative infinity.

When determining the limit as \( x \) approaches positive infinity, we analyze the function's behavior as \( x \) increases without bound. For example, consider the function \( f(x) = \frac{1}{x} \). As \( x \) approaches infinity, the values of \( f(x) \) get closer and closer to 0, indicating that the limit is:

\[ \lim_{x \to \infty} \frac{1}{x} = 0 \]

Similarly, for the function \( g(x) = \frac{5}{x} \), we can evaluate its limit as \( x \) approaches infinity by substituting increasingly larger values for \( x \). As we plug in values like 1, 10, 100, and so forth, we observe that \( g(x) \) approaches 0:

\[ \lim_{x \to \infty} \frac{5}{x} = 0 \]

On the other hand, when considering limits as \( x \) approaches negative infinity, we examine the function's behavior as \( x \) decreases without bound. For instance, the sine function, \( h(x) = \sin(x) \), oscillates between -1 and 1. As \( x \) approaches negative infinity, the function does not settle at a specific value but continues to oscillate. Therefore, we conclude that:

\[ \lim_{x \to -\infty} \sin(x) \text{ does not exist} \]

In summary, when evaluating limits at infinity, it is essential to analyze the end behavior of the function. For rational functions like \( \frac{1}{x} \) and \( \frac{5}{x} \), the limits approach 0 as \( x \) approaches positive infinity. In contrast, oscillating functions like \( \sin(x) \) do not converge to a single value as \( x \) approaches negative infinity, leading to the conclusion that their limits do not exist. Understanding these concepts is fundamental for further studies in calculus and mathematical analysis.