# 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.

Since the exponent approaches , the quantity approaches .

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.

Differentiate using the Power Rule which states that is where .

Differentiate using the Exponential Rule which states that is where =.

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.

Since the exponent approaches , the quantity approaches .

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.

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 .

Differentiate using the Exponential Rule which states that is where =.

Move the term outside of the limit because it is constant with respect to .

Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .

Multiply by .