# Evaluate limit as t approaches 0 of 3/t-3/(t^2+t) Combine terms.
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 .
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.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Take the limit of each term.
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 .
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 each term.
Raising to any positive power yields .
Multiply by .
Multiply by .
Evaluate the limit of the denominator.
Take the limit of each term.
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 .
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Raising to any positive power yields .
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.
Differentiate the numerator and denominator.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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 .
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.
Apply the distributive property.
Combine terms.
Multiply by .
Multiply by .
Subtract from .
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.
Apply the distributive property.
Combine terms.
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by .
Evaluate the limit of the numerator and the limit of the denominator.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
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.
Take the limit of each term.
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 .
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Simplify each term.
Raising to any positive power yields .
Multiply by .
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.
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 .
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 .
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.
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 .
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 .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify the denominator.
Multiply by .     