Evaluate the Limit

Problem

(lim_x→0)(sin(x)/tan(x))

Solution

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

(lim_x→0)(sin(x)/sin(x)/cos(x))

2. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

(lim_x→0)(sin(x))⋅cos(x)/sin(x)

3. Cancel the common factor of sin(x) from the numerator and the denominator, noting that x→0 implies sin(x)≠0

(lim_x→0)(cos(x))

4. Evaluate the limit by substituting x=0 into the remaining expression.

cos(0)=1

Final Answer

(lim_x→0)(sin(x)/tan(x))=1

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