Evaluate the Limit

Problem

(lim_x→0)(sin(5*x)/sin(2*x))

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in sin(0)/sin(0)=0/0

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

3. Differentiate the numerator sin(5*x) using the chain rule to get 5*cos(5*x)

4. Differentiate the denominator sin(2*x) using the chain rule to get 2*cos(2*x)

5. Rewrite the limit using these derivatives.

6. Evaluate the new limit by substituting x=0 into the simplified expression.

7. Simplify the result using the fact that cos(0)=1

Final Answer

(lim_x→0)(sin(5*x)/sin(2*x))=5/2

Want more problems? Check here!