Answer
See the explanation
Work Step by Step
The expressions \( \lim _{x \rightarrow \infty} f(x)=L \) and \( \lim _{x \rightarrow -\infty} f(x)=L \) represent the behavior of a function \( f(x) \) as the independent variable \( x \) approaches positive infinity and negative infinity, respectively.
When \( \lim _{x \rightarrow \infty} f(x)=L \), it means that as \( x \) becomes arbitrarily large, the values of \( f(x) \) approach a finite limit \( L \). In other words, the function \( f(x) \) settles down or stabilizes around the value \( L \) as \( x \) goes towards positive infinity.
Similarly, when \( \lim _{x \rightarrow -\infty} f(x)=L \), it indicates that as \( x \) becomes increasingly negative, the values of \( f(x) \) approach the same finite limit \( L \). The function \( f(x) \) stabilizes or converges to \( L \) as \( x \) approaches negative infinity.
**Example:**
Consider the function \( f(x) = \frac{1}{x} \).
1. \( \lim _{x \rightarrow \infty} \frac{1}{x} = 0 \): As \( x \) approaches positive infinity, the function \( \frac{1}{x} \) approaches 0. This means that as \( x \) becomes very large, the values of \( \frac{1}{x} \) get closer and closer to 0.
2. \( \lim _{x \rightarrow -\infty} \frac{1}{x} = 0 \): As \( x \) approaches negative infinity, the function \( \frac{1}{x} \) also approaches 0. This implies that as \( x \) becomes very negative, the values of \( \frac{1}{x} \) approach 0.
In both cases, the function \( f(x) = \frac{1}{x} \) approaches the same finite limit \( L = 0 \) as \( x \) goes towards positive or negative infinity. Therefore, the limits \( \lim _{x \rightarrow \infty} f(x) \) and \( \lim _{x \rightarrow -\infty} f(x) \) are both equal to 0.
