# Calculus Examples

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

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.

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

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

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

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

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

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

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

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

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of by plugging in for .

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Simplify the numerator.

Use the power rule to combine exponents.

Add and to get .

Raise to the power of to get .

Subtract from to get .

Simplify the denominator.

Rewrite.

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

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

Evaluate the limit of the numerator.

Take the limit of each term.

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

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

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

Evaluate the limits by plugging in for all occurrences of .

Evaluate the limit of by plugging in for .

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Simplify the answer.

Simplify each term.

Remove parentheses around .

Raise to the power of to get .

Multiply by to get .

Subtract from to get .

Subtract from to get .

Evaluate the limit of the denominator.

Take the limit of each term.

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

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

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

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

Evaluate the limits by plugging in for all occurrences of .

Evaluate the limit of by plugging in for .

Evaluate the limit of by plugging in for .

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Simplify the answer.

Simplify each term.

Remove parentheses around .

Raise to the power of to get .

Remove parentheses around .

Raise to the power of to get .

Multiply by to get .

Add and to get .

Subtract from to get .

Subtract from to get .

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

Undefined

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

Undefined

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

Undefined

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.

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

Differentiate using the Power Rule which states that is where .

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 to get .

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

Add and to get .

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

Differentiate using the Power Rule which states that is where .

Differentiate using the Power Rule which states that is where .

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 to get .

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

Add and to get .

Rewrite.

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

Differentiate using the Power Rule which states that is where .

Differentiate using the Power Rule which states that is where .

Evaluate .

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

Add and to get .

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

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

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

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

Evaluate the limit of by plugging in for .

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Use the power rule to combine exponents.

Add and to get .

Raise to the power of to get .

Multiply by to get .

Add and to get .

Subtract from to get .