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

Math
Apply L’Hospital’s rule.
Tap for more steps…
Evaluate the limit of the numerator and the limit of the denominator.
Tap for more steps…
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Tap for more steps…
Multiply the argument of the limit by the conjugate.
Expand using the FOIL Method.
Tap for more steps…
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Combine the opposite terms in .
Tap for more steps…
Reorder the factors in the terms and .
Add and .
Add and .
Simplify each term.
Tap for more steps…
Multiply .
Tap for more steps…
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
Tap for more steps…
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Tap for more steps…
Cancel the common factor.
Divide by .
Simplify.
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Tap for more steps…
Move .
Use the power rule to combine exponents.
Add and .
Take the limit of each term.
Tap for more steps…
Reorder and .
The limit at infinity of a polynomial whose leading coefficient is negative is negative infinity.
Take the limit of each term.
Tap for more steps…
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.
Tap for more steps…
Differentiate the numerator and denominator.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Tap for more steps…
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.
Tap for more steps…
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.
Tap for more steps…
Multiply and .
Reorder terms.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Tap for more steps…
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 .
Tap for more steps…
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.
Tap for more steps…
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the limit argument.
Tap for more steps…
Multiply the numerator by the reciprocal of the denominator.
Combine factors.
Tap for more steps…
Multiply by .
Multiply and .
Apply L’Hospital’s rule.
Tap for more steps…
Evaluate the limit of the numerator and the limit of the denominator.
Tap for more steps…
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Tap for more steps…
Multiply the argument of the limit by the conjugate.
Expand using the FOIL Method.
Tap for more steps…
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Combine the opposite terms in .
Tap for more steps…
Reorder the factors in the terms and .
Add and .
Add and .
Simplify each term.
Tap for more steps…
Multiply by by adding the exponents.
Tap for more steps…
Move .
Multiply by .
Multiply .
Tap for more steps…
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
Tap for more steps…
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Tap for more steps…
Cancel the common factor.
Divide by .
Simplify.
Multiply by by adding the exponents.
Tap for more steps…
Move .
Multiply by .
Tap for more steps…
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
Evaluate the limit of the denominator.
Tap for more steps…
Take the limit of each term.
Tap for more steps…
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 .
Simplify the answer.
Tap for more steps…
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.
Tap for more steps…
Differentiate the numerator and denominator.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Tap for more steps…
Use to rewrite as .
Multiply by by adding the exponents.
Tap for more steps…
Move .
Multiply by .
Tap for more steps…
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.
Add and .
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.
Tap for more steps…
Multiply by .
Subtract from .
Combine and .
Combine and .
Multiply by .
Factor out of .
Cancel the common factors.
Tap for more steps…
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 .
Add and .
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 .
Add and .
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.
Tap for more steps…
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Move to the denominator using the negative exponent rule .
Simplify.
Tap for more steps…
Apply the distributive property.
Apply the distributive property.
Combine terms.
Tap for more steps…
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.
Tap for more steps…
Move .
Multiply by .
Tap for more steps…
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.
Add and .
Factor out of .
Cancel the common factors.
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Combine and .
Factor out of .
Cancel the common factors.
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
Subtract from .
Convert fractional exponents to radicals.
Tap for more steps…
Rewrite as .
Rewrite as .
Rewrite as .
Combine terms.
Tap for more steps…
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the limit argument.
Tap for more steps…
Multiply the numerator by the reciprocal of the denominator.
Combine factors.
Tap for more steps…
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Combine and .
Combine and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Cancel the common factor of and .
Tap for more steps…
Factor out of .
Cancel the common factors.
Tap for more steps…
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)

Do you need help with math?

Try our mobile app

Our app allows students to get instant step-by-step solutions to all kinds of math troubles.

Our MATH EXPERTS

JackHudson

Charlie Trom

Lucy Evel

Scroll to top