Evaluate the Limit

Problem

(lim_x→0)(sin(x)/(2*x2−x))

Solution

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

sin(0)/(2*(0)2−0)=0/0

Since the result is an indeterminate form, we must simplify the expression or use L'Hôpital's Rule.

2. Factor the denominator to isolate the variable x

sin(x)/(x*(2*x−1))

3. Rewrite the limit as a product of two separate limits using the product rule for limits.

(lim_x→0)(sin(x)/x⋅1/(2*x−1))

4. Apply the fundamental trigonometric limit identity, which states that (lim_x→0)(sin(x)/x)=1

(lim_x→0)(sin(x)/x)=1

5. Evaluate the second part of the product by substituting x=0 into the remaining fraction.

(lim_x→0)(1/(2*x−1))=1/(2*(0)−1)=−1

6. Multiply the results of the two limits to find the final value.

1⋅(−1)=−1

Final Answer

(lim_x→0)(sin(x)/(2*x2−x))=−1

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