Understanding the Behavior and Limit of the Sinc Function: Analyzing the Expression sinx/x

sinx/x

The expression sinx/x is a common mathematical function known as the sinc function

The expression sinx/x is a common mathematical function known as the sinc function. It is defined for all values of x, except at x = 0 where it is undefined.

To explain the behavior of sinx/x, let’s break it down into two cases:

Case 1: When x is not equal to 0
For any non-zero value of x, we can evaluate sinx/x easily. The term sinx represents the sine of x, which can take any value between -1 and 1. Dividing sinx by x simply scales the result based on the value of x. So, as x varies, sinx/x will give a value between -1/x and 1/x.

Case 2: When x is equal to 0
At x = 0, the expression sinx/x becomes indeterminate. This is because the denominator (x) becomes zero, resulting in division by zero which is meaningless in mathematics.

To analyze the limit of sinx/x as x approaches 0, we can use the concept of limits. Taking the limit as x approaches 0, we can use L’Hôpital’s rule to evaluate it.

Using L’Hôpital’s rule, we differentiate the numerator and denominator separately. The derivative of sinx with respect to x is cosx, and the derivative of x with respect to x is 1. So, the limit of sinx/x as x approaches 0 becomes the limit of cosx/1 as x approaches 0, which is equal to cos(0)/1 = 1/1 = 1.

Hence, the limit of sinx/x as x approaches 0 is equal to 1.

In summary, the expression sinx/x represents the sinc function, which is defined for all values of x except at x = 0. For non-zero values of x, sinx/x scales the sine function based on the value of x. As x approaches 0, the limit of sinx/x is 1.

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