Calculus Examples

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.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
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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Rewrite.
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 .
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 .
Since is constant with respect to , the derivative of with respect to is .
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.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
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...
Rewrite.
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 .
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 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.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
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...
Rewrite.
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 .
Since is constant with respect to , the derivative of with respect to is .
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 .
Multiply by .
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.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
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...
Rewrite.
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 .
Differentiate using the Power Rule which states that is where .
Multiply by .
Split the limit using the Limits Quotient Rule on the limit as approaches .
Evaluate the limits by plugging in the value for the variable.
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
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