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

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Recall that horizontal asymptotes are found by evaluating the limits of the function as \(x\) approaches \(\infty\) and \(-\infty\).

Set up the limit for \(x \to \infty\): \(\lim_{x \to \infty} x \tan\left(\frac{1}{x}\right)\).

To evaluate this limit, consider the substitution \(t = \frac{1}{x}\), so as \(x \to \infty\), \(t \to 0^+\), and rewrite the limit as \(\lim_{t \to 0^+} \frac{\tan(t)}{t^{-1}} = \lim_{t \to 0^+} \frac{\tan(t)}{t} \cdot t\).

Use the fact that \(\lim_{t \to 0} \frac{\tan(t)}{t} = 1\) to simplify the expression and analyze the behavior of the limit.

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.

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