Evaluate the Limit ( limit as x approaches 0 of tan(5x))/x

Problem

(lim_x→0)(tan(5*x)/x)

Solution

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

tan(5⋅0)/0=0/0

2. Rewrite the expression using the trigonometric identity tan(θ)=sin(θ)/cos(θ)

(lim_x→0)(sin(5*x)/(x*cos(5*x)))

3. Separate the limit into two parts to utilize the fundamental trigonometric limit (lim_θ→0)(sin(θ)/θ)=1

(lim_x→0)(sin(5*x)/x⋅1/cos(5*x))

4. Multiply and divide by 5 to match the argument of the sine function.

(lim_x→0)(5⋅sin(5*x)/(5*x)⋅1/cos(5*x))

5. Apply the limit laws and the fundamental limit (lim_5*x→0)(sin(5*x)/(5*x))=1

5⋅1⋅1/cos(0)

6. Evaluate the remaining trigonometric term where cos(0)=1

5⋅1⋅1=5

Final Answer

(lim_x→0)(tan(5*x)/x)=5

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