Understanding and Evaluating the Limit as x Approaches 0 of sin(x)/x in Calculus

limit as x approaches 0: sinx/x

sin(x)/x is also 0

To find the limit as x approaches 0 of sin(x)/x, we can use a standard trigonometric limit identities.

The limit formula for sin(x)/x as x approaches 0 is an essential limit in calculus. It helps us understand the behavior of the sine function as x approaches zero.

To evaluate the limit, we can analyze the behavior of sin(x)/x as x gets closer and closer to 0 from both the left and right sides.

First, let’s consider the limit as x approaches 0 from the left side (x < 0): As x approaches 0 from the left side, sin(x) is negative because the sine function is negative in the third and fourth quadrants of the unit circle. However, since x is also negative, the negative sign cancels out, and sin(x)/x remains positive. Therefore, as x approaches 0 from the left side, sin(x)/x approaches 0. Next, let's consider the limit as x approaches 0 from the right side (x > 0):
As x approaches 0 from the right side, sin(x) is positive because the sine function is positive in the first and second quadrants of the unit circle. Here, both x and sin(x) are positive, so sin(x)/x remains positive. Therefore, as x approaches 0 from the right side, sin(x)/x approaches 0 again.

Since the limit as x approaches 0 from both sides is the same, we can conclude that the overall limit as x approaches 0 of sin(x)/x is also 0.

In mathematical notation,
lim(x->0) sin(x)/x = 0.

This limit has important applications in calculus, especially when dealing with derivatives and trigonometric functions. It is also used to define the derivative of the sine function itself as the slope of the tangent line to the unit circle at the point (0,1).

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