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Calculus Examples
Step 1
Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.1.2
Evaluate the limit of the numerator.
Step 1.1.2.1
Evaluate the limit.
Step 1.1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.2.1.2
Evaluate the limit of which is constant as approaches .
Step 1.1.2.1.3
Move the limit inside the trig function because cosine is continuous.
Step 1.1.2.1.4
Move the term outside of the limit because it is constant with respect to .
Step 1.1.2.2
Evaluate the limit of by plugging in for .
Step 1.1.2.3
Simplify the answer.
Step 1.1.2.3.1
Simplify each term.
Step 1.1.2.3.1.1
Multiply by .
Step 1.1.2.3.1.2
The exact value of is .
Step 1.1.2.3.1.3
Multiply by .
Step 1.1.2.3.2
Subtract from .
Step 1.1.3
Evaluate the limit of the denominator.
Step 1.1.3.1
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 1.1.3.2
Move the limit inside the trig function because sine is continuous.
Step 1.1.3.3
Evaluate the limits by plugging in for all occurrences of .
Step 1.1.3.3.1
Evaluate the limit of by plugging in for .
Step 1.1.3.3.2
Evaluate the limit of by plugging in for .
Step 1.1.3.4
Simplify the answer.
Step 1.1.3.4.1
The exact value of is .
Step 1.1.3.4.2
Multiply by .
Step 1.1.3.4.3
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.3.5
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 1.2
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 1.3
Find the derivative of the numerator and denominator.
Step 1.3.1
Differentiate the numerator and denominator.
Step 1.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.4
Evaluate .
Step 1.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.4.2
Differentiate using the chain rule, which states that is where and .
Step 1.3.4.2.1
To apply the Chain Rule, set as .
Step 1.3.4.2.2
The derivative of with respect to is .
Step 1.3.4.2.3
Replace all occurrences of with .
Step 1.3.4.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.4.4
Differentiate using the Power Rule which states that is where .
Step 1.3.4.5
Multiply by .
Step 1.3.4.6
Multiply by .
Step 1.3.4.7
Multiply by .
Step 1.3.5
Add and .
Step 1.3.6
Differentiate using the Product Rule which states that is where and .
Step 1.3.7
The derivative of with respect to is .
Step 1.3.8
Differentiate using the Power Rule which states that is where .
Step 1.3.9
Multiply by .
Step 2
Move the term outside of the limit because it is constant with respect to .
Step 3
Step 3.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 3.1.1
Take the limit of the numerator and the limit of the denominator.
Step 3.1.2
Evaluate the limit of the numerator.
Step 3.1.2.1
Evaluate the limit.
Step 3.1.2.1.1
Move the limit inside the trig function because sine is continuous.
Step 3.1.2.1.2
Move the term outside of the limit because it is constant with respect to .
Step 3.1.2.2
Evaluate the limit of by plugging in for .
Step 3.1.2.3
Simplify the answer.
Step 3.1.2.3.1
Multiply by .
Step 3.1.2.3.2
The exact value of is .
Step 3.1.3
Evaluate the limit of the denominator.
Step 3.1.3.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 3.1.3.2
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 3.1.3.3
Move the limit inside the trig function because cosine is continuous.
Step 3.1.3.4
Move the limit inside the trig function because sine is continuous.
Step 3.1.3.5
Evaluate the limits by plugging in for all occurrences of .
Step 3.1.3.5.1
Evaluate the limit of by plugging in for .
Step 3.1.3.5.2
Evaluate the limit of by plugging in for .
Step 3.1.3.5.3
Evaluate the limit of by plugging in for .
Step 3.1.3.6
Simplify the answer.
Step 3.1.3.6.1
Simplify each term.
Step 3.1.3.6.1.1
The exact value of is .
Step 3.1.3.6.1.2
Multiply by .
Step 3.1.3.6.1.3
The exact value of is .
Step 3.1.3.6.2
Add and .
Step 3.1.3.6.3
The expression contains a division by . The expression is undefined.
Undefined
Step 3.1.3.7
The expression contains a division by . The expression is undefined.
Undefined
Step 3.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 3.2
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 3.3
Find the derivative of the numerator and denominator.
Step 3.3.1
Differentiate the numerator and denominator.
Step 3.3.2
Differentiate using the chain rule, which states that is where and .
Step 3.3.2.1
To apply the Chain Rule, set as .
Step 3.3.2.2
The derivative of with respect to is .
Step 3.3.2.3
Replace all occurrences of with .
Step 3.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.4
Differentiate using the Power Rule which states that is where .
Step 3.3.5
Multiply by .
Step 3.3.6
Move to the left of .
Step 3.3.7
By the Sum Rule, the derivative of with respect to is .
Step 3.3.8
Evaluate .
Step 3.3.8.1
Differentiate using the Product Rule which states that is where and .
Step 3.3.8.2
The derivative of with respect to is .
Step 3.3.8.3
Differentiate using the Power Rule which states that is where .
Step 3.3.8.4
Multiply by .
Step 3.3.9
The derivative of with respect to is .
Step 3.3.10
Simplify.
Step 3.3.10.1
Add and .
Step 3.3.10.2
Reorder terms.
Step 4
Step 4.1
Move the term outside of the limit because it is constant with respect to .
Step 4.2
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 4.3
Move the limit inside the trig function because cosine is continuous.
Step 4.4
Move the term outside of the limit because it is constant with respect to .
Step 4.5
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 4.6
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 4.7
Move the limit inside the trig function because sine is continuous.
Step 4.8
Move the term outside of the limit because it is constant with respect to .
Step 4.9
Move the limit inside the trig function because cosine is continuous.
Step 5
Step 5.1
Evaluate the limit of by plugging in for .
Step 5.2
Evaluate the limit of by plugging in for .
Step 5.3
Evaluate the limit of by plugging in for .
Step 5.4
Evaluate the limit of by plugging in for .
Step 6
Step 6.1
Cancel the common factor of .
Step 6.1.1
Factor out of .
Step 6.1.2
Factor out of .
Step 6.1.3
Cancel the common factor.
Step 6.1.4
Rewrite the expression.
Step 6.2
Multiply by .
Step 6.3
Multiply by .
Step 6.4
Combine and .
Step 6.5
The exact value of is .
Step 6.6
Simplify the denominator.
Step 6.6.1
The exact value of is .
Step 6.6.2
Add and .
Step 6.7
Multiply by .
Step 6.8
Divide by .