Evaluate the Limit ( limit as x approaches 0 of tan(4x))/x

Problem

(lim_x→0)(tan(4*x)/x)

Solution

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

tan(4*(0))/0=0/0

2. Apply the fundamental trigonometric limit identity, which states that (lim_u→0)(sin(u)/u)=1

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

(lim_x→0)(sin(4*x)/(x*cos(4*x)))

4. Manipulate the expression to create the form sin(4*x)/(4*x) by multiplying the numerator and denominator by 4

(lim_x→0)(4)⋅sin(4*x)/(4*x)⋅1/cos(4*x)

5. Evaluate the limit by applying the limit laws and substituting x=0 into the remaining terms.

4⋅1⋅1/cos(0)=4⋅1⋅1=4

Final Answer

(lim_x→0)(tan(4*x)/x)=4

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