Evaluate the Limit

Problem

(lim_x→0)((sin(x)*cos(4*x))/(x+x*cos(5*x)))

Solution

1. Factor the denominator by taking out the common term x

(sin(x)*cos(4*x))/(x*(1+cos(5*x)))

2. Separate the expression into a product of two limits using the limit laws.

(lim_x→0)(sin(x)/x⋅cos(4*x)/(1+cos(5*x)))

3. Apply the special limit rule (lim_x→0)(sin(x)/x)=1

(lim_x→0)(sin(x)/x)=1

4. Evaluate the second part of the limit by substituting x=0 into the cosine functions.

(lim_x→0)(cos(4*x)/(1+cos(5*x)))=cos(0)/(1+cos(0))

5. Simplify the numerical values using the fact that cos(0)=1

1/(1+1)=1/2

6. Multiply the results of the two limits together.

1⋅1/2=1/2

Final Answer

(lim_x→0)((sin(x)*cos(4*x))/(x+x*cos(5*x)))=1/2

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