Evaluate the Limit limit as x approaches 0 of (cos(2x))/x

Problem

(lim_x→0)(cos(2*x)/x)

Solution

1. Analyze the behavior of the numerator as x approaches 0

(lim_x→0)(cos(2*x))=cos(0)=1

2. Analyze the behavior of the denominator as x approaches 0

(lim_x→0)(x)=0

3. Determine the limit type by observing that the expression takes the form of a non-zero constant divided by zero.

1/0

4. Evaluate the one-sided limits to check for convergence. As x approaches 0 from the right (x→0, the expression cos(2*x)/x approaches positive infinity.

(lim_x→0)(cos(2*x)/x)=∞

5. Evaluate the left-hand limit. As x approaches 0 from the left (x→0, the expression cos(2*x)/x approaches negative infinity.

(lim_x→0)(cos(2*x)/x)=−∞

6. Conclude that since the one-sided limits are not equal and do not approach a finite value, the limit does not exist.

Final Answer

(lim_x→0)(cos(2*x)/x)=Does Not Exist

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