Evaluate Using L’Hospital’s Rule limit as theta approaches pi/2 of (1-sin(theta))/(1+cos(6theta))

Math
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
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Take the limit of each term.
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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 .
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Simplify the answer.
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Simplify each term.
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The exact value of is .
Multiply by .
Subtract from .
Evaluate the limit of the denominator.
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Take the limit of each term.
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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.
Move the term outside of the limit because it is constant with respect to .
Evaluate the limits by plugging in for all occurrences of .
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Simplify the answer.
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Simplify each term.
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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Remove full rotations of until the angle is between and .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
The exact value of is .
Multiply by .
Subtract from .
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.
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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 .
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Since is constant with respect to , the derivative of with respect to is .
The derivative of with respect to is .
Subtract from .
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 .
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Differentiate using the chain rule, which states that is where and .
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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 .
Subtract from .
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Tap for more steps…
Take the limit of each term.
Tap for more steps…
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 .
Simplify the answer.
Tap for more steps…
The exact value of is .
Multiply by .
Evaluate the limit of the denominator.
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Take the limit of each term.
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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.
Tap for more steps…
Cancel the common factor of .
Tap for more steps…
Factor out of .
Cancel the common factor.
Rewrite the expression.
Remove full rotations of until the angle is between and .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
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.
Find the derivative of the numerator and denominator.
Tap for more steps…
Differentiate the numerator and denominator.
Since is constant with respect to , the derivative of with respect to is .
The derivative of with respect to is .
Multiply by .
Multiply by .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
Tap for more steps…
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 .
Multiply by .
Differentiate using the Power Rule which states that is where .
Multiply by .
Take the limit of each term.
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Split the limit using the Limits Quotient Rule on the limit as approaches .
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 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 .
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Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Simplify the answer.
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Separate fractions.
Rewrite as a product.
Write as a fraction with denominator .
Simplify.
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Divide by .
Convert from to .
Move the negative in front of the fraction.
The exact value of is .
Multiply by .
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Remove full rotations of until the angle is between and .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the second quadrant.
The exact value of is .
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Evaluate Using L’Hospital’s Rule limit as theta approaches pi/2 of (1-sin(theta))/(1+cos(6theta))

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