89. Use limits to find horizontal asymptotes for each function.
a. y = x tan(1/x)

Verified step by step guidance

1. Recall that horizontal asymptotes are found by evaluating the limits of the function as \(x\) approaches \(\infty\) and \(-\infty\).
2. Set up the limit for \(x \to \infty\): \(\lim_{x \to \infty} x \tan\left(\frac{1}{x}\right)\).
3. To evaluate this limit, consider the substitution \(t = \frac{1}{x}\), so as \(x \to \infty\), \(t \to 0^+\), and rewrite the limit as \(\lim_{t \to 0^+} \frac{\tan(t)}{t^{-1}} = \lim_{t \to 0^+} \frac{\tan(t)}{t} \cdot t\).
4. Use the fact that \(\lim_{t \to 0} \frac{\tan(t)}{t} = 1\) to simplify the expression and analyze the behavior of the limit.
5. Repeat the process for \(x \to -\infty\) by considering \(t = \frac{1}{x} \to 0^-\) and evaluate \(\lim_{x \to -\infty} x \tan\left(\frac{1}{x}\right)\) similarly to find the horizontal asymptote on the left side.

Key Concepts

Limits at Infinity

Limits at infinity describe the behavior of a function as the input variable grows very large or very small. They help determine the end behavior of functions, which is essential for identifying horizontal asymptotes by evaluating the limit of the function as x approaches infinity or negative infinity.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph of a function approaches as x tends to infinity or negative infinity. It is found by calculating the limit of the function as x approaches ±∞. If the limit exists and is finite, that value is the horizontal asymptote.

Behavior of Trigonometric Functions Near Zero

Understanding how trigonometric functions behave near zero is crucial, especially for expressions like tan(1/x) as x approaches infinity. Since 1/x approaches zero, knowing that tan(θ) ≈ θ for small θ helps simplify the limit and find the horizontal asymptote.