Evaluate the Limit ( limit as x approaches 0 of sin(x))/x

Problem

(lim_x→0)(sin(x)/x)

Solution

1. Identify the form of the limit by substituting x=0 into the expression.

2. Evaluate the numerator and denominator, which results in sin(0)=0 and 0 giving the indeterminate form 0/0

3. Apply L'Hôpital's Rule, which states that if a limit results in 0/0 the limit is equal to the limit of the derivatives of the numerator and denominator.

4. Differentiate the numerator sin(x) to get cos(x) and the denominator x to get 1

5. Substitute x=0 into the new expression cos(x)/1

6. Calculate the final value using cos(0)=1

Final Answer

(lim_x→0)(sin(x)/x)=1

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