# Calculus Examples

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.

Rewrite.

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 .

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 .

Since is constant with respect to , the derivative of with respect to is .

Add and .

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.

Rewrite.

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 .

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 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.

Rewrite.

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 .

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 .

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.

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 limit of which is constant as approaches .

Evaluate the limit of which is constant as approaches .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.