Evaluate limit as t approaches 0 of 3/t-3/(t^2+t)

Math
Combine terms.
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To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Combine.
Reorder the factors of .
Combine the numerators over the common denominator.
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
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Take the limit of each term.
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Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the term outside of the limit because it is constant with respect to .
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the exponent from outside the limit using the Limits Power Rule.
Move the term outside of the limit because it is constant with respect to .
Evaluate the limits by plugging in for all occurrences of .
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Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Simplify the answer.
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Simplify each term.
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Raising to any positive power yields .
Add and .
Multiply by .
Multiply by .
Add and .
Evaluate the limit of the denominator.
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Take the limit of each term.
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Split the limit using the Product of Limits Rule on the limit as approaches .
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the exponent from outside the limit using the Limits Power Rule.
Evaluate the limits by plugging in for all occurrences of .
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Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Simplify the answer.
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Raising to any positive power yields .
Add and .
Multiply by .
The expression contains a division by The expression is undefined.
Undefined
The expression contains a division by The expression is undefined.
Undefined
The expression contains a division by The expression is undefined.
Undefined
Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
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Differentiate the numerator and denominator.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Differentiate using the Power Rule which states that is where .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Simplify.
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Apply the distributive property.
Combine terms.
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Multiply by .
Multiply by .
Subtract from .
Add and .
Differentiate using the Product Rule which states that is where and .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Differentiate using the Power Rule which states that is where .
Differentiate using the Power Rule which states that is where .
Multiply by .
Simplify.
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Apply the distributive property.
Combine terms.
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Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
Add and .
Add and .
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
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Move the term outside of the limit because it is constant with respect to .
Evaluate the limit of by plugging in for .
Multiply by .
Evaluate the limit of the denominator.
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Take the limit of each term.
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Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the term outside of the limit because it is constant with respect to .
Move the exponent from outside the limit using the Limits Power Rule.
Move the term outside of the limit because it is constant with respect to .
Evaluate the limits by plugging in for all occurrences of .
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Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Simplify the answer.
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Simplify each term.
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Raising to any positive power yields .
Multiply by .
Multiply by .
Add and .
The expression contains a division by The expression is undefined.
Undefined
The expression contains a division by The expression is undefined.
Undefined
The expression contains a division by The expression is undefined.
Undefined
Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
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Differentiate the numerator and denominator.
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Take the limit of each term.
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Split the limit using the Limits Quotient Rule on the limit as approaches .
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the term outside of the limit because it is constant with respect to .
Evaluate the limits by plugging in for all occurrences of .
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .
Simplify the answer.
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Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify the denominator.
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Multiply by .
Add and .
Divide by .
Evaluate limit as t approaches 0 of 3/t-3/(t^2+t)

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