Evaluate the Limit ( limit as x approaches 0 of tan(3x))/x

Problem

(lim_x→0)(tan(3*x)/x)

Solution

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

tan(3*(0))/0=0/0

2. Apply the fundamental trigonometric limit (lim_θ→0)(sin(θ)/θ)=1 by rewriting the tangent function.

tan(3*x)/x=sin(3*x)/(x*cos(3*x))

3. Manipulate the expression to match the argument of the sine function by multiplying the numerator and denominator by 3

(3*sin(3*x))/(3*x*cos(3*x))

4. Separate the limit into the product of two simpler limits using limit properties.

(lim_x→0)((3*sin(3*x))/(3*x))⋅(lim_x→0)(1/cos(3*x))

5. Evaluate each limit individually, noting that as x→0 3*x→0

3*(1)⋅1/cos(0)

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

3⋅1=3

Final Answer

(lim_x→0)(tan(3*x)/x)=3

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