Evaluate the Limit

Problem

(lim_x→0)(tan(2*x)/sin(5*x))

Solution

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

2. 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.

3. Differentiate the numerator tan(2*x) using the chain rule to get 2*sec2(2*x)

4. Differentiate the denominator sin(5*x) using the chain rule to get 5*cos(5*x)

5. Substitute x=0 into the new expression (2*sec2(2*x))/(5*cos(5*x))

6. Evaluate the trigonometric functions at zero, noting that sec(0)=1 and cos(0)=1

Final Answer

(lim_x→0)(tan(2*x)/sin(5*x))=2/5

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