Evaluate the limit. [tex] \lim _{x \rightarrow \infty} \frac{4 \sin x}{x}=[?][/tex]
Answer (1)
Recognize that − 1 ≤ sin x ≤ 1 .
Multiply by 4 and divide by x to get x − 4 ≤ x 4 s i n x ≤ x 4 .
Take the limit as x → ∞ and apply the Squeeze Theorem.
Conclude that lim x → ∞ x 4 s i n x = 0 .
Explanation
Problem Analysis We are asked to find the limit of the function x 4 s i n x as x approaches infinity.
Sine Function Boundedness We know that the sine function, sin x , oscillates between -1 and 1, i.e., − 1 ≤ sin x ≤ 1 .
Multiplying by a Constant Multiplying the inequality by 4, we get − 4 ≤ 4 sin x ≤ 4 .
Dividing by x Dividing by x (since x approaches infinity, we can assume 0"> x > 0 ), we have x − 4 ≤ x 4 s i n x ≤ x 4 .
Applying the Limit Now, we take the limit as x approaches infinity for all parts of the inequality: x → ∞ lim x − 4 ≤ x → ∞ lim x 4 sin x ≤ x → ∞ lim x 4
Evaluating the Limits We know that lim x → ∞ x − 4 = 0 and lim x → ∞ x 4 = 0 .
Applying the Squeeze Theorem By the Squeeze Theorem, since the function x 4 s i n x is squeezed between two functions that both approach 0 as x approaches infinity, we have x → ∞ lim x 4 sin x = 0
Final Answer Therefore, the limit of the given function as x approaches infinity is 0.
Examples
In signal processing, the function x s i n x , known as the sinc function, appears frequently. Understanding its behavior as x approaches infinity is crucial for analyzing the long-term behavior of signals. For instance, when designing filters, engineers need to ensure that the filter's response diminishes to zero as the frequency becomes very large, preventing unwanted noise from affecting the signal. The Squeeze Theorem helps confirm that such functions indeed approach zero, ensuring the filter's effectiveness.
