Calculus Examples

Evaluate Using L'Hospital's Rule limit as x approaches pi/2 of (cos(x))/(1-sin(x))
Step 1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.2
Evaluate the limit of the numerator.
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Step 1.2.1
Move the limit inside the trig function because cosine is continuous.
Step 1.2.2
Evaluate the limit of by plugging in for .
Step 1.2.3
The exact value of is .
Step 1.3
Evaluate the limit of the denominator.
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Step 1.3.1
Evaluate the limit.
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Step 1.3.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.3.1.2
Evaluate the limit of which is constant as approaches .
Step 1.3.1.3
Move the limit inside the trig function because sine is continuous.
Step 1.3.2
Evaluate the limit of by plugging in for .
Step 1.3.3
Simplify the answer.
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Step 1.3.3.1
Simplify each term.
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Step 1.3.3.1.1
The exact value of is .
Step 1.3.3.1.2
Multiply by .
Step 1.3.3.2
Subtract from .
Step 1.3.3.3
The expression contains a division by . The expression is undefined.
Undefined
Step 1.3.4
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
Find the derivative of the numerator and denominator.
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Step 3.1
Differentiate the numerator and denominator.
Step 3.2
The derivative of with respect to is .
Step 3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5
Evaluate .
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Step 3.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.2
The derivative of with respect to is .
Step 3.6
Subtract from .
Step 4
Dividing two negative values results in a positive value.
Step 5
Convert from to .
Step 6
Consider the left sided limit.
Step 7
As the values approach from the left, the function values increase without bound.
Step 8
Consider the right sided limit.
Step 9
As the values approach from the right, the function values decrease without bound.
Step 10
Since the left sided and right sided limits are not equal, the limit does not exist.