Evaluate the Limit limit as x approaches 0 of 1/x-1/(x^2+x)

Problem

(lim_x→0)(1/x−1/(x2+x))

Solution

1. Factor the denominator of the second fraction to identify a common denominator.

1/(x2+x)=1/(x*(x+1))

2. Find a common denominator for the two fractions, which is x*(x+1)

1/x⋅(x+1)/(x+1)=(x+1)/(x*(x+1))

3. Combine the fractions into a single expression over the common denominator.

(x+1)/(x*(x+1))−1/(x*(x+1))=(x+1−1)/(x*(x+1))

4. Simplify the numerator by subtracting the constants.

x/(x*(x+1))

5. Cancel the common factor of x from the numerator and the denominator, noting that x≠0 as it approaches the limit.

1/(x+1)

6. Evaluate the limit by substituting x=0 into the simplified expression.

1/(0+1)=1

Final Answer

(lim_x→0)(1/x−1/(x2+x))=1

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