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

Problem

(lim_t→0)(1/t−1/(t2+t))

Solution

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

1/(t2+t)=1/(t*(t+1))

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

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

3. Combine the fractions into a single expression.

(t+1−1)/(t*(t+1))=t/(t*(t+1))

4. Simplify the expression by canceling the common factor of t from the numerator and the denominator.

t/(t*(t+1))=1/(t+1)

5. Evaluate the limit by substituting t=0 into the simplified expression.

(lim_t→0)(1/(t+1))=1/(0+1)=1

Final Answer

(lim_t→0)(1/t−1/(t2+t))=1

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