Evaluate the Limit limit as x approaches 0 of cos(x+sin(x))

Problem

(lim_x→0)(cos(x+sin(x)))

Solution

1. Identify the type of limit. Since the function ƒ(x)=cos(x+sin(x)) is a composition of continuous functions (cosine, sine, and addition), it is continuous everywhere in its domain.

2. Apply the direct substitution property for continuous functions. This allows the limit to be evaluated by substituting the value x=0 directly into the expression.

(lim_x→0)(cos(x+sin(x)))=cos(0+sin(0))

3. Evaluate the inner trigonometric function. We know that sin(0)=0

cos(0+0)=cos(0)

4. Simplify the final expression. We know that cos(0)=1

cos(0)=1

Final Answer

(lim_x→0)(cos(x+sin(x)))=1

Want more problems? Check here!