Final answer:
The limit of sinx/x equals 1 as x approaches zero because, by the properties of the unit circle, the arc length (sine of the angle) and the tangent line (the angle itself) become nearly identical as the angle gets very small, resulting in their ratio approaching 1.
Step-by-step explanation:
The question of why the limit of sinx/x equals 1 as x approaches zero is a classic problem in calculus related to the concept of limits. The intuition behind this limit involves the unit circle and the properties of sine as an angle approaches zero. On the unit circle, the length of the arc approaches the length of the tangent line to the circle as the angle approaches zero, corresponding to the sine of the angle and the angle itself when measured in radians, respectively.
Thus, as the angle (x in this case) gets infinitesimally small, the sine of the angle and the angle become nearly equal, making their ratio approach 1.More formally, this limit can be proven using L'Hôpital's Rule or by constructing a geometric proof involving squeezing the value of sin(x) between two functions that both have the same limit as x approaches zero.
The mathematical expression of this is such that as long as we perform the same operation on both sides of the equals sign, we maintain an equality. Since any fraction with the same quantity in the numerator and the denominator equals 1, this is why the aforementioned limit results in the value of 1.