The line \(y=2\) is a horizontal asymptote for the function \(f(x)=2+1/x\) because the graph of the function gets closer to the line as \(x\) gets larger.įor other functions, the values \(f(x)\) may not approach a finite number but instead may become larger for all values of \(x\) as they get larger. For this function, we say “\(f(x)\) approaches two as \(x\) goes to infinity,” and we write \(f(x)→2\) as \(x→∞\).
For example, for the function \(f(x)=2+1/x\), the values \(1/x\) become closer and closer to zero for all values of \(x\) as they get larger and larger. For some functions, the values of \(f(x)\) approach a finite number.
To determine the behavior of a function \(f\) as the inputs approach infinity, we look at the values \(f(x)\) as the inputs, \(x\), become larger.
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