# Calculus Examples

Step 1

Step 1.1

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

Step 1.2

Evaluate the limit of the numerator.

Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as approaches .

Step 1.2.2

Move the exponent from outside the limit using the Limits Power Rule.

Step 1.2.3

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

Step 1.2.4

Evaluate the limit of which is constant as approaches .

Step 1.2.5

Evaluate the limits by plugging in for all occurrences of .

Step 1.2.5.1

Evaluate the limit of by plugging in for .

Step 1.2.5.2

Evaluate the limit of by plugging in for .

Step 1.2.6

Simplify the answer.

Step 1.2.6.1

Simplify each term.

Step 1.2.6.1.1

Raise to the power of .

Step 1.2.6.1.2

Multiply by .

Step 1.2.6.1.3

Multiply by .

Step 1.2.6.2

Subtract from .

Step 1.2.6.3

Subtract from .

Step 1.3

Evaluate the limit of the denominator.

Step 1.3.1

Split the limit using the Sum of Limits Rule on the limit as approaches .

Step 1.3.2

Move the exponent from outside the limit using the Limits Power Rule.

Step 1.3.3

Move the exponent from outside the limit using the Limits Power Rule.

Step 1.3.4

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

Step 1.3.5

Evaluate the limit of which is constant as approaches .

Step 1.3.6

Evaluate the limits by plugging in for all occurrences of .

Step 1.3.6.1

Evaluate the limit of by plugging in for .

Step 1.3.6.2

Evaluate the limit of by plugging in for .

Step 1.3.6.3

Evaluate the limit of by plugging in for .

Step 1.3.7

Simplify the answer.

Step 1.3.7.1

Simplify each term.

Step 1.3.7.1.1

Raise to the power of .

Step 1.3.7.1.2

Raise to the power of .

Step 1.3.7.1.3

Multiply by .

Step 1.3.7.1.4

Multiply by .

Step 1.3.7.2

Add and .

Step 1.3.7.3

Subtract from .

Step 1.3.7.4

Subtract from .

Step 1.3.7.5

The expression contains a division by . The expression is undefined.

Undefined

Step 1.3.8

The expression contains a division by . The expression is undefined.

Undefined

Step 1.4

The expression contains a division by . The expression 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

By the Sum Rule, the derivative of with respect to is .

Step 3.3

Differentiate using the Power Rule which states that is where .

Step 3.4

Evaluate .

Step 3.4.1

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

Step 3.4.2

Differentiate using the Power Rule which states that is where .

Step 3.4.3

Multiply by .

Step 3.5

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

Step 3.6

Add and .

Step 3.7

By the Sum Rule, the derivative of with respect to is .

Step 3.8

Differentiate using the Power Rule which states that is where .

Step 3.9

Differentiate using the Power Rule which states that is where .

Step 3.10

Evaluate .

Step 3.10.1

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

Step 3.10.2

Differentiate using the Power Rule which states that is where .

Step 3.10.3

Multiply by .

Step 3.11

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

Step 3.12

Add and .

Step 4

Split the limit using the Limits Quotient Rule on the limit as approaches .

Step 5

Split the limit using the Sum of Limits Rule on the limit as approaches .

Step 6

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

Step 7

Move the exponent from outside the limit using the Limits Power Rule.

Step 8

Evaluate the limit of which is constant as approaches .

Step 9

Split the limit using the Sum of Limits Rule on the limit as approaches .

Step 10

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

Step 11

Move the exponent from outside the limit using the Limits Power Rule.

Step 12

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

Step 13

Evaluate the limit of which is constant as approaches .

Step 14

Step 14.1

Evaluate the limit of by plugging in for .

Step 14.2

Evaluate the limit of by plugging in for .

Step 14.3

Evaluate the limit of by plugging in for .

Step 15

Step 15.1

Simplify the numerator.

Step 15.1.1

Multiply by by adding the exponents.

Step 15.1.1.1

Multiply by .

Step 15.1.1.1.1

Raise to the power of .

Step 15.1.1.1.2

Use the power rule to combine exponents.

Step 15.1.1.2

Add and .

Step 15.1.2

Raise to the power of .

Step 15.1.3

Multiply by .

Step 15.1.4

Subtract from .

Step 15.2

Simplify the denominator.

Step 15.2.1

Multiply by by adding the exponents.

Step 15.2.1.1

Multiply by .

Step 15.2.1.1.1

Raise to the power of .

Step 15.2.1.1.2

Use the power rule to combine exponents.

Step 15.2.1.2

Add and .

Step 15.2.2

Raise to the power of .

Step 15.2.3

Multiply by .

Step 15.2.4

Multiply by .

Step 15.2.5

Add and .

Step 15.2.6

Subtract from .