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

Problem

(lim_a→0)(sin(a)/a)

Solution

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

2. Observe that sin(0)=0 which results in the indeterminate form 0/0

3. Apply L'Hôpital's Rule, which states that for indeterminate forms of 0/0 the limit is equal to the limit of the derivatives of the numerator and denominator.

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

5. Evaluate the new limit as a approaches 0

(lim_a→0)(cos(a)/1)

6. Substitute a=0 into the simplified expression.

cos(0)=1

Final Answer

(lim_a→0)(sin(a)/a)=1

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