Final answer:
The limit of the transcendental function as x approaches zero, lim₁→⃠ (sin⁷x)/x, is determined by noting that lim₁→⃠ (sin(x))/x = 1, and adjusting for the seventh power of sine, resulting in a final answer of 1.
Step-by-step explanation:
The student is asking to find the limit of the transcendental function as x approaches zero, specifically lim₁→⃠ (sin⁷x)/x. To answer this, we can apply a well-known limit, which is lim₁→⃠ (sin(x))/x = 1. This basic limit is often used with functions that involve sin(x) over x as x approaches zero.
In this case, however, the function is (sin(x))^7 over x. To resolve this, we can rewrite the expression as (sin(x)/x) · (sin(x))^6. As x approaches zero, (sin(x)/x) approaches 1, and (sin(x))^6 approaches 0^6, which is also 1. Therefore, the limit of the original function as x approaches zero is 1 · 1^6, which simply equals 1.
Remember to always recognize and apply the basic trigonometric limits to more complex functions involving trigonometry and variables approaching a limit.