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The line y = 4 is a horizontal asymptote to the right of the graph of a function f. What is the limit of f as x approaches infinity?

Question

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Solution

Understanding the Concept of Horizontal Asymptotes

A horizontal asymptote indicates the behavior of a function as the variable approaches infinity. When we say that the line y=4 y = 4 is a horizontal asymptote to the right of the graph of a function f f , it implies that as x x becomes very large (approaching infinity), the value of the function f(x) f(x) approaches the value of the asymptote.

Evaluating the Limit

  1. Statement of the Limit Definition: The limit of f(x) f(x) as x x approaches infinity can be expressed mathematically as: limxf(x) \lim_{x \to \infty} f(x)

  2. Asymptotic Behavior: Given that the line y=4 y = 4 is a horizontal asymptote, we know: limxf(x)=4 \lim_{x \to \infty} f(x) = 4

Final Answer

Thus, the limit of f f as x x approaches infinity is:

limxf(x)=4 \lim_{x \to \infty} f(x) = 4

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