# Calculus Examples

Step 1

Step 1.1

Take the limit of the numerator and the limit of the denominator.

Step 1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

Step 1.3

Since the exponent approaches , the quantity approaches .

Step 1.4

Infinity divided by infinity is undefined.

Undefined

Step 2

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.

Step 3

Step 3.1

Differentiate the numerator and denominator.

Step 3.2

Differentiate using the Power Rule which states that is where .

Step 3.3

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

Step 4

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

Step 5

Step 5.1

Evaluate the limit of the numerator and the limit of the denominator.

Step 5.1.1

Take the limit of the numerator and the limit of the denominator.

Step 5.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

Step 5.1.3

Since the exponent approaches , the quantity approaches .

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

Step 5.2

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.

Step 5.3

Find the derivative of the numerator and denominator.

Step 5.3.1

Differentiate the numerator and denominator.

Step 5.3.2

Differentiate using the Power Rule which states that is where .

Step 5.3.3

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

Step 6

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

Step 7

Multiply by .