Evaluate the Limit

Problem

(lim_x→0)((x2−x+sin(x))/(2*x))

Solution

1. Identify the form of the limit by substituting x=0 into the expression.

2. Evaluate the numerator and denominator at x=0 0−0+sin(0)=0 and 2(0) = 0$.

3. Recognize that the limit is in the indeterminate form 0/0 which allows the use of L'Hôpital's Rule.

4. Apply L'Hôpital's Rule by taking the derivative of the numerator and the derivative of the denominator separately.

5. Differentiate the numerator: (d(x2)−x+sin(x))/d(x)=2*x−1+cos(x)

6. Differentiate the denominator: (d(2)*x)/d(x)=2

7. Rewrite the limit using these derivatives.

(lim_x→0)((2*x−1+cos(x))/2)

8. Substitute x=0 into the new expression to find the limit value.

(2*(0)−1+cos(0))/2

9. Simplify the result using cos(0)=1

(0−1+1)/2=0/2=0

Final Answer

(lim_x→0)((x2−x+sin(x))/(2*x))=0

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