# 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.

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.

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 limit inside the trig function because cosine is continuous.

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

Move the limit inside the trig function because cosine is continuous.

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

Move the limit inside the trig function because cosine is continuous.

Evaluate the limit of by plugging in for .

Evaluate the limit of by plugging in for .

Evaluate the limit of by plugging in for .

Simplify the numerator.

The exact value of is .

Multiply by to get .

Multiply by to get .

The exact value of is .

Multiply by to get .

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

Move the limit inside the trig function because sine is continuous.

Move the limit inside the trig function because sine is continuous.

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 .

Simplify the answer.

Simplify each term.

The exact value of is .

Multiply by to get .

Multiply by to get .

The exact value of is .

Multiply by to get .

Add and 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 limit inside the trig function because sine is continuous.

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 .

Simplify the answer.

Simplify each term.

The exact value of is .

Multiply by to get .

Add and 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

Find the derivative of the numerator and denominator.

Rewrite.

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

Evaluate .

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

The derivative of with respect to is .

Evaluate .

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

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

The derivative of with respect to is .

Replace all occurrences of with .

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 .

Move to the left of the expression .

Multiply by to get .

Multiply by to get .

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 .

The derivative of with respect to is .

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

Move the limit inside the trig function because cosine is continuous.

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

Move the limit inside the trig function because cosine is continuous.

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 .

Simplify the answer.

Simplify each term.

The exact value of is .

Multiply by to get .

Multiply by to get .

The exact value of is .

Multiply by 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 limit inside the trig function because cosine is continuous.

Evaluate the limits by plugging in for all occurrences of .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of by plugging in for .

Simplify the answer.

Simplify each term.

The exact value of is .

Multiply by 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

Find the derivative of the numerator and denominator.

Rewrite.

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

Evaluate .

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

The derivative of with respect to is .

Multiply by to get .

Evaluate .

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

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

The derivative of with respect to is .

Replace all occurrences of with .

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 .

Multiply by to get .

Multiply by to get .

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

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

Evaluate .

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

The derivative of with respect to is .

Multiply by to get .

Multiply by to get .

Add and to get .

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

Move the limit inside the trig function because sine is continuous.

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

Move the limit inside the trig function because sine is continuous.

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 .

Simplify the answer.

Simplify each term.

The exact value of is .

Multiply by to get .

Multiply by to get .

The exact value of is .

Multiply by to get .

Add and to get .

Evaluate the limit of the denominator.

Move the limit inside the trig function because sine is continuous.

Evaluate the limit of by plugging in for .

The exact value of is .

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

Undefined

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

Undefined

Find the derivative of the numerator and denominator.

Rewrite.

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

Evaluate .

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

The derivative of with respect to is .

Evaluate .

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

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

The derivative of with respect to is .

Replace all occurrences of with .

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 .

Move to the left of the expression .

Multiply by to get .

Multiply by to get .

The derivative of with respect to is .

Rewrite.

The derivative of with respect to is .

Move the limit inside the trig function because cosine is continuous.

Evaluate the limit of by plugging in for .

The exact value of is .

Divide by to get .