# Calculus Examples

Evaluate the limit of the numerator and the limit of the denominator.
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.
Find the derivative of the numerator and denominator.
Rewrite.
Differentiate using the Power Rule which states that is where .
Differentiate using the Exponential Rule which states that is where =.
Evaluate the limit of the numerator and the limit of the denominator.
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.
Find the derivative of the numerator and denominator.
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 =.
Take the limit of each term.
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 .