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

Problem

(lim_x→0)(x/tan(x))

Solution

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

2. Evaluate the numerator and denominator at x=0 which gives 0/tan(0)=0/0

3. Apply L'Hôpital's Rule, which states that if a limit results in an indeterminate form 0/0 the limit is equal to the limit of the derivatives of the numerator and denominator.

4. Differentiate the numerator x with respect to x to get 1

5. Differentiate the denominator tan(x) with respect to x to get sec2(x)

6. Rewrite the limit using these derivatives.

(lim_x→0)(1/sec2(x))

7. Substitute x=0 into the new expression.

1/sec2(0)

8. Simplify the result using the fact that sec(0)=1

1/1=1

Final Answer

(lim_x→0)(x/tan(x))=1

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