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 .

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.

Multiply by .

Multiply by .

The exact value of is .

Multiply by .

Add and .

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

Move the limit inside the trig function because tangent 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.

Multiply by .

Multiply by .

The exact value of is .

Multiply by .

Add and .

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.

Differentiate the numerator and denominator.

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 .

Differentiate using the Power Rule which states that is where .

Multiply by .

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 .

Move to the left of .

Multiply by .

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 .

Differentiate using the Power Rule which states that is where .

Multiply by .

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 .

Move to the left of .

Multiply by .

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 .

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.

Multiply by .

The exact value of is .

Multiply by .

Subtract from .

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 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 limit inside the trig function because secant 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 which is constant as approaches .

Evaluate the limit of by plugging in for .

Simplify the answer.

Reorder and .

Factor out of .

Factor out of .

Factor out of .

Apply pythagorean identity.

Multiply by .

The exact value of is .

Raising to any positive power yields .

Multiply by .

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.

Differentiate the numerator and denominator.

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 .

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 .

Multiply by .

Multiply by .

Add and .

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 .

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

To apply the Chain Rule, set as .

Differentiate using the Power Rule which states that is where .

Replace all occurrences of with .

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 .

Move to the left of .

Remove parentheses.

Multiply by .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Remove parentheses.

Subtract from .

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

Evaluate the limit of the numerator.

Take the limit of each term.

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 limit of by plugging in for .

Simplify the answer.

Multiply by .

The exact value of is .

Multiply by .

Evaluate the limit of the denominator.

Take the limit of each term.

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

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

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

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

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

Move the limit inside the trig function because tangent 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.

Multiply by .

The exact value of is .

One to any power is one.

Multiply by .

Multiply by .

The exact value of is .

Multiply by .

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.

Differentiate the numerator and denominator.

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 .

Remove parentheses.

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

Multiply by .

Differentiate using the Power Rule which states that is where .

Multiply by .

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

Differentiate using the Product Rule which states that is where and .

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 .

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Add and .

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

Remove parentheses.

Move to the left of .

Differentiate using the Power Rule which states that is where .

Multiply by .

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

To apply the Chain Rule, set as .

Differentiate using the Power Rule which states that is where .

Replace all occurrences of with .

Remove parentheses.

Remove parentheses.

Move to the left of .

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 .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

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

Multiply by .

Differentiate using the Power Rule which states that is where .

Multiply by .

Simplify.

Apply the distributive property.

Combine terms.

Multiply by .

Multiply by .

Remove parentheses.

Reorder terms.

Split the limit using the Limits Quotient 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 .

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 .

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

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

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

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 limit inside the trig function because tangent is continuous.

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

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 limit inside the trig function because secant is continuous.

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 by plugging in for .

Evaluate the limit of by plugging in for .

Simplify the numerator.

Multiply by .

The exact value of is .

Simplify the denominator.

Multiply by .

The exact value of is .

One to any power is one.

Multiply by .

Multiply by .

The exact value of is .

Raising to any positive power yields .

Multiply by .

Multiply by .

The exact value of is .

One to any power is one.

Multiply by .

Subtract from .

Multiply by .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Evaluate Using L’Hospital’s Rule limit as x approaches 0 of (4x-sin(4x))/(4x-tan(4x))