Limits and Infinity


Find the limits in Exercises 37–46.


sin x
lim ------------- ( If you have a grapher, try graphing
x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .

Verified step by step guidance

1. Identify the function for which you need to find the limit: \( \lim_{{x \to \infty}} \frac{{\sin x}}{{|x|}} \).
2. Understand the behavior of \( \sin x \): The sine function oscillates between -1 and 1 for all real numbers.
3. Consider the behavior of \(|x|\) as \(x\) approaches infinity: The absolute value of \(x\) increases without bound.
4. Analyze the fraction \(\frac{{\sin x}}{{|x|}}\): Since \(\sin x\) is bounded and \(|x|\) grows indefinitely, the fraction's numerator remains between -1 and 1 while the denominator increases.
5. Conclude that as \(x\) approaches infinity, the fraction \(\frac{{\sin x}}{{|x|}}\) approaches 0, because a bounded numerator divided by an unbounded denominator tends to zero.