College Math Placement Practice Test

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What is the result of the limit lim(x→0) (sin(x)/x)?

0
1
Infinity
-1

The correct answer is: 1

The limit of $\lim_{x \to 0} \frac{\sin(x)}{x}$ has a fundamental significance in calculus and is often used in various applications, including determining derivatives of trigonometric functions. As $x$ approaches 0, both the numerator $\sin(x)$ and the denominator $x$ approach 0, creating an indeterminate form of $\frac{0}{0}$. To evaluate this limit, one can use several methods such as L'Hôpital's Rule, Taylor series expansion, or geometric interpretations involving the unit circle. Using L'Hôpital's Rule involves differentiating the numerator and denominator separately. The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $x$ is 1. Thus, applying L'Hôpital's Rule gives: \[ \lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac{\cos(x)}{1} \] As $x$ approaches 0, $\cos(x)$ approaches $\cos(0) = 1$. Therefore, the limit evaluates to 1.