Exploring the Limit of sin(1/x) / (1/x) as x Approaches 0
In this article, we will delve into the evaluation of a specific limit involving the sine function and explore its behavior as the variable approaches zero. This exploration will include rewriting the expression, applying the Squeeze Theorem, and ultimately determining the limit. Along the way, we will also discuss related concepts in calculus and their practical implications.
Introduction to the Problem
The limit we are interested in is given by:
[ lim_{x to 0} frac{sinleft(frac{1}{x}right)}{frac{1}{x}}. ]At first glance, this limit may seem daunting, but through careful analysis and application of key theorems in calculus, we can simplify the expression and arrive at a definitive answer. Let's start by rewriting the expression in a more manageable form.
Rewriting the Expression
We begin by making a substitution:
[ y frac{1}{x}. ]As ( x ) approaches 0, ( y ) approaches infinity. This substitution transforms the original limit into:
[ lim_{x to 0} frac{sinleft(frac{1}{x}right)}{frac{1}{x}} lim_{y to infty} frac{sin(y)}{y}. ]Now, we need to analyze the behavior of ( frac{sin(y)}{y} ) as ( y ) approaches infinity. The sine function oscillates between -1 and 1 for all values of ( y ). Therefore, we can write the inequality:
[ -1 leq sin(y) leq 1. ]Applying the Squeeze Theorem
Dividing the whole inequality by ( y ), which is positive as ( y ) approaches infinity, we get:
[ -frac{1}{y} leq frac{sin(y)}{y} leq frac{1}{y}. ]As ( y ) approaches infinity, both ( -frac{1}{y} ) and ( frac{1}{y} ) approach 0. By the Squeeze Theorem, we can conclude:
[ lim_{y to infty} frac{sin(y)}{y} 0. ]Therefore, we have:
[ lim_{x to 0} frac{sinleft(frac{1}{x}right)}{frac{1}{x}} 0. ]The final answer is:
[ boxed{0} ]
Related Concepts and Applications
Understanding this limit is crucial in various fields of engineering and physics where oscillating functions are common. For instance, in signal processing, the behavior of signals near zero frequency can be analyzed using similar techniques.
Moreover, this problem illustrates the power and utility of the Squeeze Theorem, a fundamental tool in calculus. The theorem allows us to bound the values of a function and make definitive conclusions about its limits, even when direct computation is challenging.
Conclusion
In conclusion, the limit of ( frac{sinleft(frac{1}{x}right)}{frac{1}{x}} ) as ( x ) approaches 0 is indeed 0. This result is not only mathematically interesting but also has practical applications in fields such as engineering and physics.