Question

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{\sin 2 x}{x}$$

Short answer

The limit of the function \(f(x)=\frac{\sin 2x}{x}\) as \( x \rightarrow \infty \) and as \( x \rightarrow -\infty \) is 0. The horizontal asymptote of the function is y = 0.

Step-by-step solution

Understanding the function

The function is in the form \(f(x)=\frac{\sin 2x}{x}\), where the numerator is an oscillating sin function with values ranging between -1 and 1, and the denominator is x which becomes very large as x approaches ±∞.

Limit as x Approaches ∞

As \( x \rightarrow \infty \), the sine function will continue to oscillate between -1 and 1, and since it's being divided by x (which is becoming increasingly larger), the resulting values will become increasingly smaller, contriving to 0. Thus, the limit as \( x \rightarrow \infty \) for the function \(f(x)=\frac{\sin 2x}{x}\) is \( \lim _{x \rightarrow\infty} f(x) = 0\).

Limit as x Approaches -∞

In a similar manner, as \( x \rightarrow -\infty\), the sine function will oscillate between -1 and 1 while the denominator becomes increasingly larger (in negative direction), making the resulting fraction skew to 0. Thus, the limit as \( x \rightarrow -\infty \) is \( \lim _{x \rightarrow-\infty} f(x)= 0\).

Identifying Horizontal Asymptotes

The horizontal asymptotes of a function are the values that the function approaches as x approaches ±∞. As found in Steps 2 and 3, this function approaches 0 as x approaches both ∞ and -∞. Therefore, the function \(f(x)=\frac{\sin 2x}{x}\) has a horizontal asymptote at y = 0.