Evaluate the Limit

Problem

(lim_x→0)(cos(x−1)/sin(x))

Solution

1. Evaluate the limit by direct substitution of x=0 into the expression.

2. Calculate the value of the numerator by substituting 0 for x

cos(0−1)=cos(−1)

3. Apply the even function identity cos(−θ)=cos(θ) to simplify the numerator.

cos(−1)=cos(1)

4. Calculate the value of the denominator by substituting 0 for x

sin(0)=0

5. Analyze the behavior of the fraction as the denominator approaches 0 while the numerator approaches a non-zero constant cos(1)≈0.5403

6. Determine the limit behavior. Since the numerator is a positive constant and the denominator sin(x) changes sign at x=0 the one-sided limits will differ.

(lim_x→0)(cos(x−1)/sin(x))=∞

(lim_x→0)(cos(x−1)/sin(x))=−∞

7. Conclude that because the left-hand and right-hand limits are not equal, the two-sided limit does not exist.

Final Answer

(lim_x→0)(cos(x−1)/sin(x))=Does Not Exist

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