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

Problem

(lim_x→0)(tan(x)/x)

Solution

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

tan(0)=0

0/0

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

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

3. Separate the expression into a product of two fractions to isolate the known special limit.

sin(x)/(x*cos(x))=sin(x)/x⋅1/cos(x)

4. Apply the limit laws to evaluate the limit of the product as 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)))

5. Evaluate each limit using the special limit (lim_x→0)(sin(x)/x)=1 and direct substitution for the cosine term.

(lim_x→0)(sin(x)/x)=1

(lim_x→0)(1/cos(x))=1/cos(0)=1

6. Multiply the results to find the final value.

1⋅1=1

Final Answer

(lim_x→0)(tan(x)/x)=1

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