Evaluating the Limit of sin(x)/x as x Approaches 0 Using L’Hôpital’s Rule

lim x->0 sinx/x

To evaluate the limit of sin(x)/x as x approaches 0, we can apply L’Hôpital’s rule, which states that if the limit of f(x)/g(x) as x approaches a specific value is of the form 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and denominator and evaluating the resulting limit

To evaluate the limit of sin(x)/x as x approaches 0, we can apply L’Hôpital’s rule, which states that if the limit of f(x)/g(x) as x approaches a specific value is of the form 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and denominator and evaluating the resulting limit.

Let’s apply L’Hôpital’s rule to the given limit:

lim x->0 sin(x)/x

Taking the derivative of sin(x) with respect to x gives us cos(x):

lim x->0 cos(x)/1

Now, substituting x = 0 into the expression, we get:

cos(0)/1 = 1/1 = 1

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

This result can be visually confirmed using the graph of y = sin(x)/x. As x approaches 0 from both positive and negative sides, the function converges to 1.

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