Evaluate the Limit ( limit as x approaches 0 of tan(2x))/x

Problem

(lim_x→0)(tan(2*x)/x)

Solution

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

tan(2*(0))/0=0/0

2. Rewrite the tangent function using the identity tan(θ)=sin(θ)/cos(θ)

(lim_x→0)(sin(2*x)/(x*cos(2*x)))

3. Apply the special limit rule (lim_u→0)(sin(u)/u)=1 by multiplying the numerator and denominator by 2 to match the argument of the sine function.

(lim_x→0)((2*sin(2*x))/(2*x*cos(2*x)))

4. Separate the limit into a product of simpler limits.

2⋅(lim_x→0)(sin(2*x)/(2*x))⋅(lim_x→0)(1/cos(2*x))

5. Evaluate each limit individually.

2⋅1⋅1/cos(0)

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

2⋅1⋅1=2

Final Answer

(lim_x→0)(tan(2*x)/x)=2

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