Explain the meaning of lim x→−∞ f(x)=10.

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1. Step 1: Understand the notation: The expression \( \lim_{x \to -\infty} f(x) = 10 \) is read as 'the limit of \( f(x) \) as \( x \) approaches negative infinity is 10.'
2. Step 2: Conceptualize the behavior: This means that as \( x \) becomes very large in the negative direction (i.e., \( x \) goes to negative infinity), the values of the function \( f(x) \) get closer and closer to 10.
3. Step 3: Visualize the graph: Imagine the graph of \( f(x) \). As you move left along the x-axis towards negative infinity, the y-values (outputs of \( f(x) \)) approach the horizontal line \( y = 10 \).
4. Step 4: Consider the horizontal asymptote: The line \( y = 10 \) can be considered a horizontal asymptote of the function \( f(x) \) as \( x \to -\infty \). This means the graph of \( f(x) \) gets closer to this line but may not necessarily touch or cross it.
5. Step 5: Relate to real-world scenarios: In practical terms, this limit could represent a situation where a quantity stabilizes at a certain value (10 in this case) as time or another variable decreases without bound.