The Limit as x Approaches 0 of sinx/x: Evaluating the Expression and Determining the Limit

limit as x approaches 0: sinx/x

To find the limit as x approaches 0 of sinx/x, we can evaluate the expression directly

To find the limit as x approaches 0 of sinx/x, we can evaluate the expression directly.

We know that sin(0) = 0, so the expression becomes sin(0)/0. However, dividing by zero is undefined, so we need to approach x = 0 from both the left and right sides to determine the limit.

Let’s evaluate the limit as x approaches 0 from the left side (x < 0). When x is negative and close to zero, sinx remains bounded between -1 and 1. So, sinx/x will also remain bounded. As x approaches 0 from the left side, sinx/x will approach a value between -1 and 1. Now, let's evaluate the limit as x approaches 0 from the right side (x > 0).

When x is positive and close to zero, sinx remains bounded between -1 and 1 similar to the left side. Again, sinx/x will remain bounded. As x approaches 0 from the right side, sinx/x will also approach a value between -1 and 1.

Since both the left and right limits approach a value between -1 and 1, we can conclude that the overall limit as x approaches 0 of sinx/x exists and is equal to this value.

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

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