Find the limit as x approaches π of (cos x +1)/(x-π)
To find the limit as x approaches π of (cos x + 1)/(x - π), you can use the concept of limits. Here's how to do it step-by-step:
Step 1: Start by substituting π into the expression for x, so you have (cos π + 1)/(π - π).
Step 2: Simplify the expression in the numerator. Since cosine of π is -1, the expression becomes (-1 + 1)/(π - π), which simplifies to 0/0.
Step 3: Realize that 0/0 is an indeterminate form, meaning it cannot be directly evaluated. In this case, we can use L'Hôpital's rule.
Step 4: Apply L'Hôpital's rule by taking the derivative of the numerator and denominator separately. The derivative of cos x is -sin x, and the derivative of x - π is 1.
Step 5: Substituting the derivatives back into the expression gives (-sin π)/(1), which simplifies to 0/1.
Step 6: The expression 0/1 equals 0.
Therefore, the limit as x approaches π of (cos x + 1)/(x - π) is 0.