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

Problem

(lim_x→0)(tan(x)/x)

Solution

1. Rewrite the tangent function using the identity tan(x)=sin(x)/cos(x)

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

2. Separate the expression into a product of two fractions to isolate the known fundamental trigonometric limit.

sin(x)/(x*cos(x))=sin(x)/x⋅1/cos(x)

3. Apply the limit product rule, which states that the limit of a product is the product of the limits.

(lim_x→0)(sin(x)/x⋅1/cos(x))=((lim_x→0)(sin(x)/x))⋅((lim_x→0)(1/cos(x)))

4. Evaluate the first limit using the fundamental limit theorem (lim_x→0)(sin(x)/x)=1

(lim_x→0)(sin(x)/x)=1

5. Evaluate the second limit by direct substitution, noting that cos(0)=1

(lim_x→0)(1/cos(x))=1/1=1

6. Multiply the results of the two limits.

1⋅1=1

Final Answer

(lim_x→0)(tan(x)/x)=1

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