Understanding L’Hôpital’s Rule to Find the Limit of sin(x)/x as x Approaches 0

limit as x approaches 0: sinx/x

To find the limit of the function f(x) = sin(x)/x as x approaches 0, we can use the concept of L’Hôpital’s Rule, which states that if both the numerator and denominator of a function approach 0 or infinity as x approaches a certain value, then the limit can be found by taking the derivative of the numerator and denominator and evaluating the new function at that given value

To find the limit of the function f(x) = sin(x)/x as x approaches 0, we can use the concept of L’Hôpital’s Rule, which states that if both the numerator and denominator of a function approach 0 or infinity as x approaches a certain value, then the limit can be found by taking the derivative of the numerator and denominator and evaluating the new function at that given value.

In this case, as x approaches 0, both sin(x) and x approach 0. Let’s apply L’Hôpital’s rule to find the limit:

1. Take the derivative of the numerator and the denominator:
The derivative of sin(x) is cos(x),
The derivative of x is 1.

2. Now, we have the new function g(x) = cos(x)/1.

3. Evaluate the new function at x = 0:
g(0) = cos(0)/1 = 1/1 = 1.

Therefore, the limit of sin(x)/x as x approaches 0 is equal to 1.

This can be interpreted as the slope of the graph of sin(x)/x at x = 0 is 1, which means that the function becomes steeper and closer to a straight line as x approaches 0.

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