Evaluate the Limit

Problem

(lim_x→0)((1−cos(2*x))/(x*tan(2*x)))

Solution

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

(1−cos(0))/(0⋅tan(0))=(1−1)/0=0/0

2. Apply trigonometric identities to rewrite the numerator and denominator.

1−cos(2*x)=2*sin2(x)

tan(2*x)=sin(2*x)/cos(2*x)

3. Substitute these identities into the limit expression.

(lim_x→0)((2*sin2(x))/(x⋅sin(2*x)/cos(2*x)))

4. Simplify the expression using the double angle identity sin(2*x)=2*sin(x)*cos(x)

(lim_x→0)((2*sin2(x)*cos(2*x))/(x⋅2*sin(x)*cos(x)))

5. Cancel common terms in the numerator and denominator.

(lim_x→0)((sin(x)*cos(2*x))/(x*cos(x)))

6. Separate the limit into known fundamental limits.

(lim_x→0)(sin(x)/x)⋅(lim_x→0)(cos(2*x)/cos(x))

7. Evaluate each limit using the property (lim_x→0)(sin(x)/x)=1 and direct substitution for the cosine terms.

1⋅cos(0)/cos(0)=1⋅1/1=1

Final Answer

(lim_x→0)((1−cos(2*x))/(x*tan(2*x)))=1

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