91Ó°ÊÓ

When analyzing a function, our goal is to understand its behavior at different points, including those that are not in its domain. In this exercise, the function we are dealing with is \( f(x) = \frac{\sin(x)}{x} \), with a focus on the point \( c = 0 \), which is outside the function's domain since dividing by zero is undefined. Analyzing a function involves evaluating its limits, which helps us determine how the function behaves as it approaches certain values.
To start this analysis, compare the values of the function as \( x \) approaches zero from both the left (negative) and the right (positive) sides. This method of function analysis, through asymmetric limits, helps clarify the behavior and tendencies of the function near the undefined point.

The sine function, represented as \( \sin(x) \), is a periodic and oscillating function crucial in trigonometry and calculus. In our case, we are examining the expression \( \frac{\sin(x)}{x} \), a well-known limit problem in calculus.
The sine function has interesting properties, such as repeating every \(2\pi\) radians and being an odd function, meaning \( \sin(-x) = -\sin(x) \). These properties justify certain behaviors of the function as \( x \) approaches zero. As \( x \) gets smaller and approaches zero, the value of \( \sin(x) \) is approximately equal to \( x \) itself, which is an essential aspect contributing to the limit of \( \frac{\sin(x)}{x} \) approaching 1.

Limit guessing involves approximating the value that a function approaches as the input value gets close to a point. In this exercise, we are trying to find the limit \( \ell = \lim_{x \to 0} \frac{\sin(x)}{x} \). By observing function values as they approach from both the left and the right, we can form a hypothesis.
When computing the function for small values near the problematic point \( c = 0 \), both from negative and positive sides, we notice a pattern: the values are very close to 1. This suggests that as \( x \rightarrow 0 \), the function values are converging towards 1, leading us to guess that \( \ell = 1 \). Thus, the limit guessing approach combines numerical exploration and pattern recognition to arrive at a plausible limit prediction.

The delta-epsilon definition is a formal method for proving the limits of a function. To say \( \lim_{x \to c} f(x) = \ell \) means that for every epsilon \( \varepsilon > 0 \), there exists a delta \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - \ell| < \varepsilon \).
In practical terms, this means that we can make \( f(x) \) as close to \( \ell \) as we desire by selecting \( x \) sufficiently close to \( c \). For our problem, we determined that choosing \( \delta = 0.1 \) ensures \( |\frac{\sin(x)}{x} - 1| < 0.01 \) for \( x \) near zero. This confirms the limit \( \ell = 1 \) with precise and controlled accuracy, fully supporting our earlier guess and specifying with mathematical rigor the limit behavior of the function.