Evaluating the Limit of sin(x)/x as x Approaches Infinity

The limit of sin(x)/x as x approaches infinity is a fundamental concept in calculus and analysis. This limit often arises in the context of understanding the behavior of oscillating functions in relation to linear functions. Let's explore this in detail.

Understanding the Oscillating Nature of sin(x)

The sine function, sin(x), oscillates between -1 and 1 for all real values of x. This characteristic oscillation means that sin(x)/x will also oscillate, but its amplitude will diminish as x increases. For large values of x, the fraction sin(x)/x will be bounded between -1/x and 1/x.

Applying the Squeeze Theorem

The Squeeze Theorem is a powerful tool for evaluating limits. We can apply this theorem to sin(x)/x as follows:

We know that sin(x) oscillates between -1 and 1 for all sub>x. Therefore, -1/x ≤ sin(x)/x ≤ 1/x. As sub>x approaches infinity, both -1/x and 1/x approach 0.

By the Squeeze Theorem, if a function is squeezed between two other functions that both approach the same limit, the function in the middle will also approach that limit. Thus, we conclude that:

limsub x → ∞ sin(x)/x 0

Exploring Additional Cases

The behavior of sin(x)/x can depend on the value that x approaches:

Case 1: x → c ≠ 0
If x approaches a non-zero constant c, the function sin(x)/x is continuous at c. Therefore, the limit is simply the value of the function at that point:

limsub x → c sin(x)/x sin(c) / c

Case 2: x → 0
When x approaches 0, we can use the power series expansion of sin(x):

sin(x) x - x3 / 3! x5 / 5! - ...

Dividing by x, we get:

sin(x) / x 1 - x2 / 3! x4 / 5! - ...

As x approaches 0, the higher-order terms approach 0. Therefore:

limsub x → 0 sin(x)/x 1

Special Cases for Different Approaches

The limit can also be evaluated for different approaches. For example:

Approaching infinity: limsub x → ∞ sin(x)/x 0 as we discussed earlier. Approaching zero: limsub x → 0 sin(x)/x 1, using the power series expansion. Approaching a ≠ 0: limsub x → a sin(x)/x sin(a) / a.

Conclusion

Understanding the limit of sin(x)/x as x approaches infinity is crucial for analyzing oscillatory behavior in mathematical functions. By leveraging the Squeeze Theorem and power series expansions, we can confidently evaluate these limits and gain deeper insights into the behavior of functions as variables approach specific values.