# Evaluate limit as t approaches infinity of ( square root of t+t^2)/(6t-t^2)

Apply L’Hospital’s rule.
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.
Multiply the argument of the limit by the conjugate.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Combine the opposite terms in .
Reorder the factors in the terms and .
Simplify each term.
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify.
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Take the limit of each term.
Reorder and .
The limit at infinity of a polynomial whose leading coefficient is negative is negative infinity.
Take the limit of each term.
Reorder and .
The limit at infinity of a polynomial whose leading coefficient is negative is negative infinity.
Infinity divided by infinity is undefined.
Undefined
Infinity divided by infinity 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 .
Use to rewrite as .
Differentiate using the Power Rule which states that is where .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Differentiate using the Power Rule which states that is where .
Rewrite the expression using the negative exponent rule .
Simplify.
Multiply and .
Reorder terms.
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 .
Reorder terms.
Rewrite as .
Combine terms.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the limit argument.
Multiply the numerator by the reciprocal of the denominator.
Combine factors.
Multiply by .
Multiply and .
Apply L’Hospital’s rule.
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.
Multiply the argument of the limit by the conjugate.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Combine the opposite terms in .
Reorder the factors in the terms and .
Simplify each term.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify.
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by .
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
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 .
As approaches for radicals, the value goes to .
The limit at infinity of a polynomial whose leading coefficient is negative is negative infinity.
Evaluate the limit of which is constant as approaches .
A non-zero constant times infinity is infinity.
Infinity times infinity is infinity.
Infinity divided by infinity is undefined.
Undefined
Infinity divided by infinity is undefined.
Undefined
Infinity divided by infinity 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 .
Use to rewrite as .
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Combine and .
Combine and .
Multiply by .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Since is constant with respect to , the derivative of with respect to is .
Use to rewrite as .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Product Rule which states that is where and .
By the Sum Rule, the derivative of with respect to is .
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 .
Since is constant with respect to , the derivative of with respect to is .
Move to the left of .
Differentiate using the Power Rule which states that is where .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Move to the denominator using the negative exponent rule .
Simplify.
Apply the distributive property.
Apply the distributive property.
Combine terms.
Multiply by .
Combine and .
Combine and .
Move to the left of .
Move to the numerator using the negative exponent rule .
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Combine and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Combine terms.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the limit argument.
Multiply the numerator by the reciprocal of the denominator.
Combine factors.
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Combine and .
Combine and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
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.
Evaluate limit as t approaches infinity of ( square root of t+t^2)/(6t-t^2)

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